ATHENA Space Telescope - DiVA...

IN DEGREE PROJECT MATHEMATICS,SECOND CYCLE, 30 CREDITS

, STOCKHOLM SWEDEN 2017

ATHENA Space Telescope:Line of Sight Control with a Hexapod in the Loop

SIMON GÖRRIES

KTH ROYAL INSTITUTE OF TECHNOLOGYSCHOOL OF ENGINEERING SCIENCES

ATHENA Space Telescope: Line of Sight Control with a Hexapod in the SIMON GÖRRIES Degree Projects in Systems Engineering (30 ECTS credits) Degree Programme in, Aerospace Engineering, (120 credits) KTH Royal Institute of Technology year 2017 Supervisor at Airbus: Thomas Ott Supervisor at KTH: Per Engvist Examiner at KTH: Per Engvist

TRITA-MAT-E 2017:71 ISRN-KTH/MAT/E--17/71--SE Royal Institute of Technology School of Engineering Sciences KTH SCI SE-100 44 Stockholm, Sweden URL: www.kth.se/sci

I

Acknowledgment

Herewith I would like to thank all those, who supported me in writing this thesis through their technical and scientific know-how as well as their personal encouragement.

In particular, I would like to thank my thesis advisors Thomas Ott at Airbus and Prof. Per Enqvist at KTH, who supported me throughout the technical work as well as the writing of this thesis. Thomas’ office door was always open whenever I ran into trouble or had a question regarding my work. He consistently allowed this thesis to be my own work but helped me to find the right direction from time to time and supported me with his know-how even during vacation periods or after office hours. Per supported me with his academic advise via email and several skype calls during my work at Airbus and with his valuable comments on this thesis report.

I would also like to thank Dr. Jens Levenhagen, who enabled me to write my thesis at the AOCS, GNC and Flight Dynamics department at Airbus Space System in Friedrichshafen, Germany, as well as the experts of the Athena project team. In particular, I would like to thank Alexander Schleicher and Uwe Schäfer who always had an open door and took the time to discuss and answer my questions regard the Athena mission as well as Harald Langenbach for his support with the implementation of a simplified actuator model.

Finally, I must express my very profound gratitude to my parents and my girlfriend for providing me with unfailing support and continuous encouragement throughout my years of study and through the process of researching and writing this thesis. This accomplishment would not have been possible without them.

Thank you.

III

Sammanfattning

Teleskopet Athena, The Advanced Telescope for High-Energy Astrophysics, är ett röntgenteleskop med en spegelmodul som väger 2000 kg monterat på en struktur med sex frihetsgrader. I dagsläget pågår en förstudie av teleskopet som en L-klass mission inom European Space Agencys Cosmic Vision 2015-2025 med planerad uppskjutning år 2028. Satelliten väger totalt 8000 kg vilket huvudsakligen fördelas mellan spegelmodulen på ena sidan och fokalplansmodulen på andra sidan av teleskopet. Tidigare har inga europeiska projekt använt sig av en liknande design och det medför utmaningar i analys och design av styrsystemet med den rörliga spegeln medräknad. Dels leder spegelns massa till tidsvarianta osäkerheter i systemets parametrar men även den hexapoda monteringsanordningen komplicerar styralgoritmerna och medför komplexa störningar i satellitens attityd.

Dessa problem behandlas i denna rapport i tre steg. Först har ett Matlab®/Simulink® bibliotek byggts upp där alla delar som krävs för att skapa en modell av en hexapod i simuleringen av styrsystemet ingår. Detta bibliotek innefattar en komplett kinematik-modell för en hexapod, en förenklad ställdonsmodell, en dynamisk modell av satellitens attityd, tillståndsbestämning samt styralgoritmer för kombinerade rörelser av satelliten och hexapoden. Därefter designas, analyseras och jämförs olika operativa scenarier. Slutligen simulerades det återkopplade styrsystemet för en representativ fallstudie som liknar Athena för en första genomförbarhetsanalys samt jämförelse av prestanda mellan olika scenarier.

Med dessa simuleringar har först de tidsvarianta osäkerheter i systemets parametrar och störningsmoment orsakade av spegelns rörelser karakteriserats. Därigenom har störningsbruset från stegkvantiseringen till hexapodens ställdon identifierats som något som potentiellt kan driva designen och därför bör undersökas närmare i projektets tidiga skeden. Därefter har genomförbarheten av ett grundläggande koncept för styrsystemet påvisats, det vill säga manövrera heaxpoden och satelliten sekventiellt i tiden. Möjliga förbättringar till detta koncept har analyserats och gett en 27% snabbare förflyttningstid mellan två observationer i det simulerade scenariot genom att hexapoden och satelliten manövreras samtidigt längst en optimerad bana för satelliten.

V

Abstract

The Advanced Telescope for High-Energy Astrophysics, Athena, is an x-ray telescope with a 2000 kg mirror module mounted on a six degree of freedom hexapod mechanism. It is currently assessed in a phase A feasibility study as L-class mission in the European Space Agency’s Cosmic Vision 2015-2025 plan with a launch foreseen in 2028. The total mass of the spacecraft is approximately 8000 kg, which is mainly distributed to the mirror module on the one side and to the focal plane module on the other side of the telescope tube. Such a design is without precedent in any European mission and imposes several challenges on analysis and design of the complex line of sight pointing control system with the moving mirror module in the loop. Not only does the moving mirror mass lead to time-variant parameter uncertainties of the system (inertia), but also does the hexapod motion complicate the guidance algorithms and induce complex disturbances onto the SC attitude dynamics.

These challenges have been approached in this thesis in three steps. First, a Matlab®/Simulink® library has been built up, including all components required to model a hexapod in pointing control simulations. This Hexapod Simulation Library includes a complete hexapod kinematic model, a simplified hexapod actuator model, a spacecraft attitude dynamic model with the hexapod in the loop, state determination algorithms with the hexapod in the loop as well as online guidance algorithms for combined spacecraft and hexapod maneuvers. Second, different operational scenarios have been designed and analyzed for comparison. Third, closed-loop pointing control simulations for a representative reference case study similar to the Athena spacecraft have been performed for first feasibility analysis and performance comparison of the different operational scenarios.

With these simulations, first, the time-variant parameter uncertainties and complex disturbance torques caused by the moving mirror mass have been characterized. Thereby, the disturbance noise induced by hexapod actuator step quantization has been identified as a potential design driver that needs to be analyzed further in early phases of the project. Second, feasibility of a baseline pointing control concept has been shown, i.e. performing hexapod and spacecraft maneuvers sequentially in time. And third, possible improvements to the baseline concept have been analyzed providing up to 27% faster transition time between two observations for the simulated scenario by performing hexapod and spacecraft maneuvers simultaneously and applying a path optimized spacecraft trajectory.

VII

Contents

List of Figures ................................................................................................................................. XI

List of Tables ................................................................................................................................ XV

List of Symbols ........................................................................................................................... XVII

List of Acronyms ........................................................................................................................... XIX

1 Introduction ............................................................................................................................... 1

1.1 Overview ..................................................................................................................................... 1

1.2 Problem Context ......................................................................................................................... 1

1.2.1 Athena Mission ................................................................................................................. 1

1.2.2 Reference Scenario ........................................................................................................... 6

1.2.3 Pointing Control Design Process ....................................................................................... 6

1.3 Thesis Objective and Methodology ............................................................................................ 7

1.4 Contributions .............................................................................................................................. 8

1.5 Thesis Outline ............................................................................................................................. 9

2 Theoretical Background ............................................................................................................ 11

2.1 Overview ................................................................................................................................... 11

2.2 Nomenclature ........................................................................................................................... 11

2.3 Classical Pointing Control System ............................................................................................. 11

2.3.1 Classical Pointing Control Components .......................................................................... 12

2.3.2 Classical Spacecraft Attitude Dynamics .......................................................................... 14

2.3.3 Classical Disturbances and Model Uncertainties ............................................................ 14

2.4 Hexapod Mechanism ................................................................................................................ 14

2.4.1 Geometry and State Definition ....................................................................................... 15

2.4.2 Position and Orientation ................................................................................................ 16

2.5 Newton-Euler Formulation for Multi-Body Dynamics .............................................................. 18

3 Pointing Control System with Hexapod in the Loop .................................................................. 21

3.1 Overview ................................................................................................................................... 21

3.2 System Description and Repointing Process ............................................................................ 21

3.3 Comparison to Classical Pointing Control System and Related Design Challenges .................. 22

3.4 Reference Coordinate Frames .................................................................................................. 24

3.4.1 Earth Centered Rotating Frame {ECR} ............................................................................ 24

3.4.2 Spacecraft Attitude Reference Frame {I} ........................................................................ 25

3.4.3 Spacecraft Body Frame {B} ............................................................................................. 26

3.4.4 Detector Frames {Di} ...................................................................................................... 27

3.4.5 Line of Sight Frame {LoS} ................................................................................................ 28

3.4.6 Hexapod Base Reference Frame {H} ............................................................................... 29

3.4.7 Hexapod Platform Reference Frame {P0} ....................................................................... 30

VIII Contents

3.4.8 Hexapod Platform Moving Frame {P} ............................................................................ 31

3.4.9 Hexapod Actuator Leg Reference Frame {Li} ................................................................. 32

3.4.10 Hexapod Local Tetrahedron Frame {Ti} ......................................................................... 33

4 Hexapod Open Loop Control Chain ........................................................................................... 35

4.1 Overview .................................................................................................................................. 35

4.2 Hexapod Inverse Kinematic ...................................................................................................... 36

4.3 Hexapod Forward Kinematic .................................................................................................... 39

4.3.1 Forward Pose Analysis ................................................................................................... 39

4.3.2 Forward Rate Analysis .................................................................................................... 48

4.3.3 Forward Acceleration Analysis ....................................................................................... 50

4.4 Actuator Simplified Substitute Model ...................................................................................... 52

5 Spacecraft Attitude Dynamics with Hexapod in the Loop .......................................................... 55

5.1 Overview .................................................................................................................................. 55

5.2 Two-Body System with Prescribed Relative Motion in Free Space ......................................... 56

5.3 Spacecraft Attitude Dynamics with Prescribed Hexapod Motion ........................................... 58

5.3.1 Hexapod Induced Inertia Variation ................................................................................ 62

5.3.2 Leg Actuator Forces ....................................................................................................... 62

6 Pointing System State Determination with Hexapod in the Loop .............................................. 65

6.1 Overview .................................................................................................................................. 65

6.2 MOA Misalignment Measurement Based Approach ............................................................... 65

6.2.1 State Determination System Overview ......................................................................... 66

6.2.2 Knowledge Errors ........................................................................................................... 67

6.2.3 On-Board Metrology ...................................................................................................... 69

6.2.4 Hexapod State Metrology .............................................................................................. 69

6.3 MAM Absolute Pose Measurement Based Approach .............................................................. 69

6.3.1 State Determination System Overview ......................................................................... 70

6.3.2 Knowledge Errors ........................................................................................................... 71

6.3.3 On-Board Metrology ...................................................................................................... 73

6.4 Comparison .............................................................................................................................. 73

7 Maneuver Guidance with Hexapod in the Loop ........................................................................ 75

7.1 Overview .................................................................................................................................. 75

7.2 Operational Flow ...................................................................................................................... 76

7.3 Line of Sight Guidance with Hexapod in the Loop ................................................................... 78

7.4 Spacecraft Attitude Trajectory ................................................................................................. 79

7.5 Hexapod Pose Trajectory ......................................................................................................... 81

7.5.1 Hexapod Actuator Domain Guidance Algorithm ........................................................... 81

7.5.2 Hexapod State Domain Guidance Algorithm ................................................................. 82

7.5.3 Comparison .................................................................................................................... 84

7.5.4 Step Command Generation ........................................................................................... 84

8 Reference Case Study ............................................................................................................... 87

8.1 Overview .................................................................................................................................. 87

Contents IX

8.2 Parametrization ........................................................................................................................ 87

8.2.1 Reference Spacecraft Parameters .................................................................................. 87

8.2.2 Re-Pointing Scenario Parameters ................................................................................... 88

8.3 Characterization of Hexapod Effects on Spacecraft Attitude Control ...................................... 89

8.3.1 Hexapod Induced Disturbance Torque ........................................................................... 89

8.3.2 Time-Variant Inertia Uncertainty ................................................................................... 91

8.4 Simulation Cases ....................................................................................................................... 91

8.5 Performance Analysis and Comparison .................................................................................... 92

9 Pointing System Design Trade-Offs ........................................................................................... 93

9.1 Overview ................................................................................................................................... 93

9.2 Operational Flow ...................................................................................................................... 93

9.3 Re-Pointing Approach ............................................................................................................... 93

10 Conclusion ................................................................................................................................ 95

10.1 Summary ................................................................................................................................... 95

10.2 Outlook ..................................................................................................................................... 96

Appendix A Derivations and Side Notes ......................................................................................... 99

A.1 Rotation Matrix Time Derivatives ............................................................................................. 99

A.2 Radial Acceleration of Platform Junction Points in Spherical Coordinates ............................ 100

A.3 LoS Reconstruction ................................................................................................................. 101

A.4 Bang Slew Bang Trajectory ..................................................................................................... 103

Appendix B Simulation Results .................................................................................................... 105

B.1 Simulation Case 1.1: Nominal Operational Flow + Classical Re-Pointing ............................... 105

B.2 Simulation Case 1.2: Enhanced Operational Flow + Classical Re-Pointing ............................. 106

B.3 Simulation Case 2.1: Nominal Operational Flow + Enhanced Re-Pointing ............................. 106

B.4 Simulation Case 2.2: Enhanced Operational Flow + Enhanced Re-Pointing ........................... 107

Glossary ...................................................................................................................................... 109

Reference Documents ................................................................................................................. 111

XI

List of Figures

FIGURE 1-1: DEFLECTION OF THE X-RAY BEAM IN WOLTER-SCHWARZSCHILD MIRROR ASSEMBLY MODULE ..............................................................................................................................................1

FIGURE 1-2: POINTING ERROR INDICES AS DEFINED IN [5] ................................................................2

FIGURE 1-3: ILLUSTRATION OF RKE AND AKE PURPOSE [5] ................................................................3

FIGURE 1-4: ATHENA SPACECRAFT DESIGN CONCEPT [5] ...................................................................4

FIGURE 1-5: POINTING GEOMETRY OF THE ATHENA SPACECRAFT [5] ...............................................4

FIGURE 1-6: POINTING CONTROL DESIGN PROCESS ...........................................................................6

FIGURE 1-7: SPIRAL MODEL FOR SOFTWARE DEVELOPMENT ............................................................8

FIGURE 2-1: (A) CLASSICAL AND (B) ADVANCED CLASSICAL POINTING CONTROL SYSTEM ............. 12

FIGURE 2-2: CLASSICAL POINTING CONTROL LOOP ......................................................................... 12

FIGURE 2-3: REACTION WHEEL PRINCIPLE [9] ................................................................................. 12

FIGURE 2-4: INTERDISCIPLINARY POINTING ERROR SOURCE REACTION WHEEL ............................ 13

FIGURE 2-5: TORQUE ON A SC DUE TO A SINGLE THRUSTER (A) AND POSSIBLE TORQUES FOR A PAIR OF THRUSTERS (B) ............................................................................................................................ 13

FIGURE 2-6: GENERAL GEOMETRY OF A 6-DOF STEWART PLATFORM IN (A) 3D VIEW AND (B) TOP VIEW [15] .......................................................................................................................................... 15

FIGURE 2-7: FREE BODY DIAGRAM OF A RIGID BODY WITH TWO CONNECTIONS TWO OTHER RIGID BODIES [16] ...................................................................................................................................... 19

FIGURE 3-1: POINTING CONTROL SYSTEM WITH HEXAPOD IN THE LOOP: BLOCK DIAGRAM ......... 22

FIGURE 3-2: COORDINATE FRAMES OVERVIEW AND TRANSFORMATIONS .................................... 24

FIGURE 3-3: SPACECRAFT ATTITUDE REFERENCE FRAME {I} AND EARTH CENTERED ROTATING FRAME {ECR} [17] ............................................................................................................................. 25

FIGURE 3-4: SPACECRAFT BODY FRAME {B} (SOLAR PANELS FOLDED) [17] .................................... 26

FIGURE 3-5: DETECTOR FRAME {DI} AND SC BODY FRAME {B} (SOLAR PANELS FOLDED)............... 27

FIGURE 3-6: LOS FRAME {LOS} AND DETECTOR FRAME {DI} (SOLAR PANELS FOLDED) .................. 28

FIGURE 3-7: HEXAPOD BASE REFERENCE FRAME {H} ...................................................................... 29

FIGURE 3-8: HEXAPOD PLATFORM REFERENCE FRAME {P0} HEXAPOD BASE REFERENCE FRAME {H} .......................................................................................................................................................... 30

FIGURE 3-9: HEXAPOD PLATFORM MOVING FRAME {P} AND HEXAPOD PLATFORM REFERENCE FRAME {P0} ....................................................................................................................................... 31

FIGURE 3-10: HEXAPOD ACTUATOR LEG REFERENCE FRAME {LI} AND HEXAPOD BASE REFERENCE FRAME {H} ........................................................................................................................................ 32

FIGURE 3-11: EXEMPLARY LOCAL TETRAHEDRON FRAME {T1} IN (A) AND GENERIC IN (B) AND HEXAPOD BASE REFERENCE FRAME {H} ........................................................................................... 33

XII List of Figures

FIGURE 4-1: HEXAPOD STATE DOMAIN AND ACTUATOR STATE DOMAIN LINKED BY HEXAPOD KINEMATIC ........................................................................................................................................ 35

FIGURE 4-2: IMPLEMENTATION OF ACTUATOR MODEL INTO SIMULATION WITH HEXAPOD KINEMATIC ........................................................................................................................................ 36

FIGURE 4-3: 6-3 GEOMETRY OF A 6-DOF STEWART PLATFORM IN (A) 3D VIEW AND (B) ILLUSTRATING THE THREE TETRAHEDRONS [24] ............................................................................. 40

FIGURE 4-4: LOCAL TETRAHEDRON FRAME (A), PROJECTION OF THE PLATFORM JUNCTION POINT ONTO THE X-AXIS OF THE LOCAL TETRAHEDRON FRAME (B) AND BASIS VECTORS OF THE PLATFORM FRAME (C) ......................................................................................................................................... 41

FIGURE 4-5: FORWARD POSE ANALYSIS COMPUTATION TIME OVER ERROR METRIC FOR DIFFERENT APPROACHES .................................................................................................................................... 47

FIGURE 4-6: LOCAL ACTUATOR REFERENCE FRAME AND AUXILIARY VARIABLES [15] .................... 50

FIGURE 4-7: KINEMATIC CONSTRAINT OF RIGID BODY MOTION [15] ............................................. 50

FIGURE 4-8: HEXAPOD LINEAR ACTUATOR DESIGN CONCEPT ......................................................... 52

FIGURE 4-9: LINEAR ACTUATOR SIMPLIFIED SUBSTITUTE MODEL................................................... 52

FIGURE 4-10: ACTUATOR HYSTERESIS CURVE OF (A) DETAILED MODEL AND (B) SIMPLIFIED SUBSTITUTE MODEL ......................................................................................................................... 53

FIGURE 4-11: ACTUATOR STEP RESPONSE OF DETAILED MODEL AND SIMPLIFIED SUBSTITUTE MODEL .............................................................................................................................................. 53

FIGURE 5-1: SYSTEM OF TWO RIGID BODIES CONNECTED BY LINEAR ACTUATOR LEGS (A) WITH AND (B) WITHOUT FLEXIBLE MODES ........................................................................................................ 55

FIGURE 5-2: SYSTEM OF TWO RIGID BODIES IN FREE SPACE ........................................................... 57

FIGURE 5-3: TWO-BODY SYSTEM FORMED BY SPACECRAFT AND HEXAPOD PLATFORM ................ 58

FIGURE 6-1: MOA MISALIGNMENT MEASUREMENT BASED STATE DETERMINATION APPROACH . 66

FIGURE 6-2: ABSOLUTE POSE MEASUREMENT BASED STATE DETERMINATION APPROACH .......... 70

FIGURE 7-1: HEXAPOD MANEUVER OVERVIEW ............................................................................... 75

FIGURE 7-2: COMPARISON OF (A) NOMINAL AND (B) ENHANCED OPERATIONAL FLOW ............... 77

FIGURE 7-3: MANEUVER TIME FOR NOMINAL AND ENHANCED OPERATIONAL FLOW ................... 77

FIGURE 7-4: SC ATTITUDE CHANGE DURING REPOINTING MANEUVER .......................................... 79

FIGURE 7-5: SC BORESIGHT TRACE FOR (A) CLASSICAL AND (B) ENHANCED RE-POINTING APPROACH SHOWN IN {I}-FRAME ....................................................................................................................... 79

FIGURE 7-6: ROTATION ANGLES OVER TIME FOR (A) CLASSICAL AND (B) ENHANCED RE-POINTING APPROACH ........................................................................................................................................ 79

FIGURE 7-7: EXEMPLARY ILLUSTRATION OF A BANG-BANG MANEUVER IN (A) AND BANG-SLEW-BANG MANEUVER IN (B) .................................................................................................................. 80

FIGURE 7-8: HEXAPOD ACTUATOR STATE DOMAIN GUIDANCE ALGORITHM FLOW CHART ........... 82

FIGURE 7-9: EXEMPLARY PLOT OF (A) ACTUATOR LENGTHS AND (B) HEXAPOD STATES OVER TIME FOR ACTUATOR STATE DOMAIN GUIDANCE .................................................................................... 82

FIGURE 7-10: HEXAPOD STATE DOMAIN GUIDANCE ALGORITHM FLOW CHART ............................ 83

https://d.docs.live.net/efcab7c25a027a49/thesis/masterThesisV2_kthVersion.docx#_Toc494715302

https://d.docs.live.net/efcab7c25a027a49/thesis/masterThesisV2_kthVersion.docx#_Toc494715303

List of Figures XIII

FIGURE 7-11: EXEMPLARY PLOT OF (A) ACTUATOR LENGTHS AND (B) HEXAPOD STATES OVER TIME FOR HEXAPOD STATE DOMAIN GUIDANCE ...................................................................................... 83

FIGURE 7-12: DIFFERENCE BETWEEN ACTUATOR LENGTHS FOR HEXAPOD STATE DOMAIN GUIDANCE COMPARED TO ACTUATOR STATE DOMAIN GUIDANCE ............................................... 84

FIGURE 7-13:EXECUTED MOTOR STEPS IN (A) AND COMMANDED ACTUATOR LENGTH VS ACTUAL ACTUATOR LENGTH IN (B) ................................................................................................................ 86

FIGURE 8-1: SC ATTITUDE AND ISM CONFIGURATION (A) BEFORE AND (B) AFTER RE-POINTING MANEUVER ....................................................................................................................................... 89

FIGURE 8-2: HEXAPOD INDUCED DISTURBANCES OVER TIME WITHOUT STEP QUANTIZATION FOR (A) NOF AND (B) EOF ........................................................................................................................ 89

FIGURE 8-3: SQUARE ROOT OF CUMULATED DISTURBANCE POWER SPECTRUM WITH STEP QUANTIZATION ................................................................................................................................ 90

FIGURE 8-4: HEXAPOD INDUCED INERTIA ERROR FOR (A) NOF AND (B) EOF .................................. 91

FIGURE 10-1: HEXAPOD STATE DOMAIN GUIDANCE ALGORITHM FLOW CHART ........................... 97

FIGURE A-1: SPHERICAL COORDINATES AT POINT P ...................................................................... 101

FIGURE A-2: ACCELERATION, VELOCITY AND DISTANCE PLOTS OVER TIME FOR (A) BANG-SLEW-BANG AND (B) BANG-BANG MANEUVER ....................................................................................... 103

FIGURE B-1: SC BORESIGHT OVER TIME FOR SIMULATION CASE 1.1 ............................................ 105




XV

List of Tables

TABLE 1-1: ATHENA SC POINTING PERFORMANCE AND KNOWLEDGE REQUIREMENTS IMPACT AND ALLOCATION [1] ..................................................................................................................................5

TABLE 6-1: COMPARISON OF DIFFERENT STATE DETERMINATION APPROACHES IN TERMS OF KNOWLEDGE ERRORS....................................................................................................................... 74

TABLE 6-2: COMPARISON OF DIFFERENT STATE DETERMINATION APPROACHES IN TERMS OF TECHNOLOGY READINESS LEVEL ...................................................................................................... 74

TABLE 7-1: COMPARISON OF NOMINAL AND ENHANCED OPERATIONAL FLOW ............................ 78

TABLE 7-2: RE-POINTING DESIGN TRADE-OFF CONSIDERATIONS.................................................... 81

TABLE 8-1: SC AND MAM MASS AND INERTIA PROPERTIES ............................................................ 87

TABLE 8-2: SC AND HEXAPOD DIMENSIONS .................................................................................... 88

TABLE 8-3: SC AND HEXAPOD CONFIGURATION BEFORE AND AFTER RE-POINTING MANEUVER .. 88

TABLE 8-4: OPERATIONAL FLOW MANEUVER START TIMES AND DURATION ................................. 89

TABLE 8-5: SIMULATION CASES OVERVIEW AND REFERENCE NUMBER ......................................... 91

TABLE 8-6: SIMULATION CASES PERFORMANCE COMPARISON ...................................................... 92

XVII

List of Symbols

Roman Letters

𝐅 Force 𝐇 Angular momentum 𝐉 Jacobian matrix 𝐋 Actuator leg vector (from base to platform junction point) 𝐌 Torque 𝐐 Linear momentum 𝐑 Rotation matrix 𝐓 Transformation matrix from Tait-Brian angles to Cartesian 𝐜 Center point 𝐟 Generalized force 𝐡 Hexapod base junction point 𝐩 Hexapod platform junction point 𝐪 Generalized momentum 𝐫 Radius 𝐮 Generalized velocity 𝐯 Translational velocity; small translational motion increment 𝐱 Generalized position; hexapod state vector 𝑓 Motor step control frequency 𝑔(𝐚, 𝐀) Element of Euclidean motion group SE(3), with 𝐚 ∈ ℝ3 being a translation

vector and 𝐀 ∈ SO(3) a rotation matrix 𝑙 Actuator leg length 𝑟𝑜𝑡x(… ) Passive rotation around the x-axis; correspondingly for y and z

Greek Letters

ϕ, θ, ψ Tait-Bryan angles, x-y-z rotation sequence 𝛀 Small rotation matrix increment 𝛕 Actuator force vector 𝛚 Rotational velocity; small rotational motion increment 𝛾 Angle between two neighboring hexapod base junction points 𝛾 Small motion within the Euclidian motion group 𝑆𝐸(3) 𝜂 Angle between the 𝑥𝐻-axis and the vertical onto the line between to

neighboring hexapod base junction points 𝜖 Small change in actuator leg length

Scripts

𝕀𝑚x𝑚 Identity matrix of size 𝑚x𝑚 𝕆𝑛x𝑚 Matrix of size 𝑛x𝑚 with all elements being zero 𝓒 Centripetal and Coriolis terms matrix 𝓖 Generalized external force 𝓘 Inertia matrix

Operators

‖…‖ Euclidean 2-Norm |… | Absolute value

XVIII List of Symbols

[… ]x Skew-symmetric matrix formed from the elements of a vector ./ Elementwise division of two matrices or vectors of the same dimension .∗ Elementwise product of two matrices or vectors of the same dimension

(Hadamard product) ∘ Scalar product × Cross product

Indices

…(𝑡𝑟𝑎) Property of the quantity … (in this case translational part)

…|H Expressed in {H}-frame

…P Object (in this case the hexapod platform P), which the quantity … belongs to

…H Relative to {H}-frame

Accents

… Homogeneous representation of vector …

… First-order time derivative … Second-order time derivative … Unit vector

XIX

List of Acronyms

ACS Attitude Control System BLWN Band-Limited White Noise CoG Center of Gravity CRP Classical Re-Pointing approach DoF Degree(s) of Freedom EOF Enhanced Operational Flow ERP Enhanced Re-Pointing approach ESA European Space Agency FMS Fixed Metering Structure FPM Focal Plane Module FPM Focal plane module HEW Half-Energy-Width IMU Inertial Measurement Unit ISM Instrument Switch Mechanism LoS Line of Sight MAM Mirror Assembly Module MOA Mirror Optical Axis MOP Mock Observation Plan NOF Nominal Operational Flow OBM On-Board Metrology OF Operational Flow PSF Point Spread Function RW Reaction Wheel SC Spacecraft SEZ Sun Exclusion Zone STR Star Tracker SVM Service Module ToO Target of Opportunity WFI Wide Field Imager X-IFU X-ray Integral Field Unit

1

1 Introduction

Introduction

1.1 Overview

This chapter provides an introduction to the context of this thesis work in Chapter 1.2. The problem definition and corresponding objectives of this thesis are defined in Chapter 1.3.It is then discussed how these objectives have been approached methodologically and which contributions are available as outcome of this work in Chapter 1.4. Finally, an overview of the structure of this report is given in Chapter 1.5.

1.2 Problem Context

1.2.1 Athena Mission

1.2.1.1 Key Factors and Mission Objectives

Athena, the Advanced Telescope for High-Energy Astrophysics is a second large (L2)-class mission in the European Space Agency’s (ESA) Cosmic Vision 2015-25 plan. It is currently assessed in a phase A2 design and feasibility study with a launch foreseen in 2018 as described in more detail in [1], [2]. It is the next generation x-ray telescope with the primary goal of mapping hot gas structures in space and determining their physical properties to search for supermassive black holes. Therefore, it is equipped with two instruments: The X-ray Integral Field Unit (X-IFU) for high-spectral resolution imaging and the Wide Field Imager (WFI) for high count rate, moderate resolution spectroscopy over a large field of view. More detailed information on the instruments can be found in [2]. The Athena telescope will incorporate the largest x-ray primary mirror ever built with an aperture diameter of 3 m [3]. This mirror assembly module (MAM) includes more than 1000 mirror modules, deflecting the x-ray beam towards the detectors as depicted in Figure 1-1. The focal length of the telescope is 12 m. The mirror modules are arranged according to the Wolter-Schwarzschild design described in further detail in [4]. The total mass of the MAM is approximately 2000 kg. The launch mass of the spacecraft is approximately 8000 kg.

x-ray beamMirror Assembly Module (MAM)

mirror modules

Figure 1-1: Deflection of the x-ray beam in Wolter-Schwarzschild mirror assembly module

The launch is foreseen in 2028 with an Ariane 6 rocket providing direct insertion into its final Lissajous orbit around the second Lagrange point of the Sun-Earth system.

2 1. Introduction

1.2.1.2 Athena System Requirements and Spacecraft Design

The Athena mission has demanding requirements in terms of availability, autonomy, agility and pointing performance, which are drivers for the system design in general and the Attitude Control System (ACS) in particular. The most important requirements are summarized and put into context hereafter. A more detailed discussion can be found in [1].

First, high operational availability for science observations of ≥85% is required with continuous observations over >50 ks. This leads to challenging requirements for all activities that are not science observations, such as slew maneuvers, instrument switching, reaction wheel off-loading, resettling times after mode-switches, etc. An exemplary sequence of observation targets has been provided by ESA to test mission design concepts against these requirements. This so called Mock Observation Plan (MOP) covers approximately one year of observations with more than 1500 slew maneuvers and a total slew angle of more than 99254 °.

Second, related to the high availability requirement is a high general re-pointing agility. This means, that the spacecraft (SC) must perform large angle slew maneuvers in a relative short amount of time. Additionally, a Sun Exclusion Zone (SEZ) of 35 ° half cone angle around the sun vector must be considered for the slew maneuver guidance. Ensuring that the telescope boresight does not enter the SEZ is a hard constraint and violation leads to mission loss because sunlight falling into the telescope leads to the destruction of the science instruments. Additionally, so called Target of Opportunity (ToO) maneuvers are even more demanding in terms of agility. These ToO are events, where the quality of potential science data degrades rapidly over time as for example supernovae or gamma ray bursts. The ToO must be reached within hours in order to collect adequate data from the event. Therefore, these unplanned maneuvers are performed with thrusters whereas nominal maneuvers are performed with Reaction Wheels (RW) only. Usually, such quick reaction times imply a costly increase in the availability and workload of the ground segment for a mission. To avoid this, it is required for Athena to autonomously plan and execute a complete maneuver between two observations including instrument switch and SEZ avoidance on-board after only providing instrument selection and quaternion based Line of Sight (LoS) orientation via telecommands.

Third, Athena is also demanding in terms of pointing precision with challenging LoS pointing performance and knowledge requirements. These requirements are formulated in terms of the following error indices as defined in [5]:

• Absolute Performance Error (APE): Difference between target parameter (e.g. LoS attitude) and actual parameter.

• Performance Drift Error (PDE): Difference between the mean values of the APE over two time intervals Δ𝑡 separated by a stability time Δ𝑡s.

• Absolute Knowledge Error (AKE): Difference between actual parameter and the known (measured or estimated) parameter.

• Relative Knowledge Error (RKE): Difference between AKE at a given time within a time interval Δ𝑡 and the mean value over the same time interval.

Figure 1-2 illustrates these definitions further, with 𝑒(𝑡) being performance or knowledge error.

t

e(t)

RPE/RKE

APE/AKEΔtΔt

PDE/KDE

Δts

Figure 1-2: Pointing error indices as defined in [5]

1. Introduction 3

The LoS RKE requirement is derived from the Half-Energy-Width (HEW) requirement of the Point Spread Function (PSF) of an observed point source. The HEW is a criterion for image quality as described in [5]. For optical instruments operating in visible light spectrum, the HEW is impacted by the LoS pointing stability over the observation, i.e. the Relative Performance Error RPE. For an x-ray instrument however, the photons arrive with a much lower frequency such that each photon can be located on the detector. Based on the knowledge of the LoS at the time of each incidence, the PSF can be reconstructed. The accuracy of the PSF reconstruction thus depends on the Relative Knowledge Error RKE over the instrument integration time of 50 ks. The AKE requirement is straight forward and determines the astrometric accuracy and thus serves as absolute reference for the RKE [1]. Figure 1-3 (a) illustrates the PSF reconstruction based on the reconstructed location of the photon incidents on the detector. Figure 1-3 (b) shows the LoS knowledge error over time with the AKE given at ACS frequency, single photon incidents at various times and the RKE over the observation time window.

(a)

APE ensures center is on detector

Reconstructed Photons

HEW

(b)

AKE

t

LoS

Kn

ow

led

ge E

rro

r Photons on Detector

ΔtACS

Observation Interval

ΔtOBS ks

0

RKE

Figure 1-3: Illustration of RKE and AKE purpose [5]

The following requirements in terms of the above mentioned error indices are specified with 95% confidence level and temporal statistical interpretation, cf. [5]:

• LoS APE ≤ 10.0 arcsec

• LoS PDE ≤ 4.0 arcsec with window time Δ𝑡 = 2.5 ks and stability time Δ𝑡s = 3.0 ks

• LoS AKE ≤ 2.0 arcsec

• LoS RKE ≤ 0.8 arcsec with window time Δ𝑡 = 50 ks

The SC design is driven by these requirements as explained hereafter. Figure 1-4 illustrates the SC design concept with the Focal Plane Module (FPM) on the one end, the Fixed Metering Structure (FSM) in the middle and the Service Module (SVM) on the other end. The FPM includes the two instruments, X-IFU and WFI. The WFI instrument is equipped with two different detectors. The MAM is mounted at the SVM end of the spacecraft via six linearly extendible actuator legs and can thus be moved in six degrees of freedom. Such a mechanism is also known as Gauge Stewart platform or hexapod and referred to in the Athena project as Instrument Switch Mechanism (ISM). As can be seen in Figure 1-4, the six actuators are attached to the FMS via six and to the MAM via three junction points. Such a structure is called a 6-3 hexapod geometry and is further discussed in Chapter 2.4. The MAM is the moving platform of the hexapod and thus both terms are used interchangeably throughout this thesis. The term SC main body is used to refer to the rest of the spacecraft, i.e. everything but the MAM. If it is clear from the context, the SC main body can also be referred to only as SC.

4 1. Introduction

Mirror Assembly Module (MAM)

Instrument Switch Mechanism (ISM)

actuators

Fixed Metering Structure (FMS)

XIFU WFI

Focal Plane Module (FPM)

Service Module (SVM)

OBM

STR

Figure 1-4: Athena spacecraft design concept [5]

The hexapod moves the mirror to, first, deflect the x-ray beam towards the different instrument detectors and adjust the focal length and, second, compensates thermal distortions of the telescope structure. Note that the Star Tracker (STR) is mounted on the moving hexapod platform and thereby is located at the end of the pointing system chain to meet the demanding LoS knowledge requirements. Additionally, the On-Board Metrology (OBM) is mounted on the opposite side of the MAM facing towards the FPM. The OBM is used to measure the effects due to thermal distortions of the telescope for compensation maneuvers with the hexapod and thereby aligning the LoS with the Mirror Optical Axis (MOA) as explained in more detail in Chapter 6. As can be seen in Figure 1-5, the LoS pointing of the Athena SC is not simply achieved by rotating the SC body-fixed axis towards the target of interest. Instead, pointing is defined by the orientation of the LoS and the orientation of the MOA. The LoS is defined by the mirror node of the MAM and the center point of the selected instrument detector. The MOA is the perpendicular w.r.t. the cross-section of the mirror. Perfect pointing, i.e. no pointing errors is achieved if:

• The LoS is pointing towards the target of interest.

• The MOA is congruent with the LoS.

The lather is required to prevent vignetting effects, degrading the image quality. Note that if MOA and LoS are congruent, then the STR also directly measures the LoS orientation in inertial space.

Target Direction

Reconstructed LoS

Line of Sight

Mirror Optical Axis

Detector

MAM

LoS AKE

STR LoS

OBM LoS

LoS APE

MOA to LoS Misalignment

Figure 1-5: Pointing geometry of the Athena spacecraft [5]

1.2.1.3 Athena Pointing Control Challenges

The LoS pointing control for the Athena mission is challenging in many different aspects. Not only the mission requirements are demanding by themselves, but the design concept with the hexapod in the loop entails additional major challenges. On the one hand, the pointing control challenges related to mission requirements are as follows: First, time-efficiency is an important aspect of every task related to any re-pointing concept. Second, autonomous and agile large angle slew maneuvers with SEZ avoidance result in the need for efficient on-board algorithms and challenging fail/safe precautions.

1. Introduction 5

Third, the demanding pointing performance and especially knowledge requirements discussed in the last chapter are in general challenging for the overall pointing control design. Thermo-elastic deformations of the FMS influence both, pointing performance and knowledge and thus need to be measured and compensated on-board to meet the demanding requirements. On the other hand, the analysis and design of a pointing control system with a hexapod in the loop is without precedent in any European mission [1]. The dynamics of such a system consist of multiple bodies because the moving MAM represents a major part of the total SC mass. Therefore, the hexapod becomes an essential and critical part of the pointing control chain. Not only does its positioning accuracy directly influence LoS pointing performance, also does its position knowledge influence the LoS knowledge with the STR being mounted on the MAM. This introduces design challenges in order to achieve the SC pointing performance and knowledge as outlined in Table 1-1. The column ‘Error Index’ in the table lists the previously introduced pointing error indices. The column ‘Impact on Top-Level Requirements’ states the top-level performance requirements that are impacted by these pointing error requirements. The ‘Pointing fraction’ gives the pointing error requirement allocation in percentage of the top-level requirement. In the case of 100%, the pointing error requirement is a top-level requirement. The ‘ACS fraction’ gives the ACS requirement allocation in percentage of the overall pointing requirement with the error value in the column to the right. The ‘Hexapod fraction’ gives the hexapod state knowledge requirement allocation in percentage of the overall pointing requirement with the error value in the column to the right. The hexapod state knowledge impacts all knowledge and performance requirements, because both, the LoS as well as the MOA pointing, are impacted by the hexapod as described in the previous chapter.

Table 1-1: Athena SC pointing performance and knowledge requirements impact and allocation [1]

Error Index *1)

Impact on Top-Level Rqmt.

Pointing Fraction

AOCS Fraction

AOCS Rqmt. Value *2)

Hexapod Fraction

Hexapod Rqmt. Value

LoS AKE Astrometric accuracy

100% ~65% 0.90 arcsec ~20% 0.28 arcsec

LoS RKE

(50 ks)

HEW of Point Spread Function

~20% ~35% 0.22 arcsec ~45% 0.28 arcsec

LoS APE Target position on focal plane

100% ~15% 1.00 arcsec ~5% 0.28 arcsec

LoS PDE

(2.5, 3.0 ks)

Dithering raster stability

100% ~20% 0.50 arcsec ~10% 0.28 arcsec

*1) All pointing rqmts are specified with the temporal statistical interpretation and LoC = 95 % and consider a OBM.

*2) The rqmt allocation excludes µVibrations, free dynamics, thermo-elastic distortions.

The table above provides an overview on the criticality of the requirements. The allocations with a high fraction as well as small allocated error values are the most critical ones and thus highlighted in bold letters. The error values allocated to the hexapod are all in bold letters because they are critical for mission success. The combination of critical requirements on the one hand and low technology readiness level due to no prior experiences with a similar system lead to the high uncertainties connected to the hexapod. This motivates the need for an early investigation of this topic.

6 1. Introduction

1.2.2 Reference Scenario

As previously mentioned, ESA provided a mock observation plan for a complete year. However, for the computationally intense ACS simulations performed within this thesis, only a single re-pointing maneuver between two observations shall be simulated. Thus, this maneuver needs to include all previously mentioned challenges. Additionally, this thesis focuses on the analysis of nominal re-pointing maneuvers using reaction wheels, not the previously mentioned ToO maneuvers using thrusters. Therefore, the reference scenario needs to include the following elements: First, a large angle SC slew maneuver with SEZ avoidance performed with reaction wheels. Second, an instrument switch and focus adjustment performed by the hexapod. Third, a measurement of the thermal distortions of the telescope structure and following compensation maneuver performed by the hexapod.

1.2.3 Pointing Control Design Process

Figure 1-6 illustrates the general process of pointing control design. In Chapter 1.3 it is then discussed how this process is influenced when adding a hexapod to the system in order to derive the thesis objectives and tasks thereafter.

The inputs to the pointing control design process are the requirements and (preliminary) system design as well as a reference scenario – all described for Athena in the previous chapters. The pointing system is then modelled mathematically and guidance trajectories are generated for the reference scenario. Based on the system model, a controller is designed and stability and robustness analysis for the closed loop are performed. Performance analysis is done with the help of closed-loop simulation results. Finally, design trade-offs can be made based on the performance results and stability analysis, leading to the next iteration of the process for the modified system design.

System Design + Requirements

Reference Scenario

Modelling + Analysis of the Pointing System

Maneuver Guidance Trajectory Generation

Guidance Trajectory

Controller Design

Controller

Closed-Loop Simulation

Simulation Results

Performance AnalysisStability + Robustness

Analysis

Design Trade-Offs

System Modells

Figure 1-6: Pointing control design process

1. Introduction 7

1.3 Thesis Objective and Methodology

The high-level objectives of this thesis are to, first, increase the technology readiness level for a pointing control system with a hexapod in the loop as described in the previous chapters. Second, show feasibility of a baseline design concept and, third, identify and analyze possible improvements to this baseline. The current baseline anticipated by the European Space Agency is to move the hexapod only when the ACS is in idle mode, to avoid interactions between the ACS and the hexapod as far as possible.

The desired tasks to fulfill these objectives have been derived by analyzing the pointing control design process introduced in the previous chapter and identifying the parts of the process that are affected by adding a hexapod to the system. The following tasks have been identified:

• Close the gaps in the existing Airbus in-house design and analysis tools to allow pointing performance simulations with a hexapod in the loop. This includes the following sub-tasks:

o Model building: Performant solutions for the complex hexapod kinematics are required for simulations and eventually on-board algorithms. Hexapod actuator effects need to be modelled to an adequate level of detail. SC attitude dynamics with the moving hexapod need to be derived.

o Motion planning for complex line of sight pointing: Trajectory generation algorithms for different combined hexapod and spacecraft maneuvers are required, taking actuator limitations into account. These algorithms need to be simple enough to be run on-board as well.

• Design operational scenarios and compare them in terms of feasibility and performance.

• Perform closed-loop simulation with the hexapod and analyze system performance.

• Based on the simulation results answer the following questions and provide inputs for the controller design:

o How does the hexapod motion interfere with the ACS (time-varying inertia and disturbances caused by moving MAM mass)?

o Can the hexapod be actuated while the ACS is active or even during a slew maneuver?

• Identify design drivers and open design points based on the above work.

The constraints implied by the existing pointing control design tool chain available at Airbus are to use Matlab®/Simulink® 2016b. The Simulink® models must be ready to be run in acceleration target mode. No additional toolboxes except Airbus in-house tools for ACS simulations shall be used.

The spiral model for software development described in [6] has been applied in order to close the gaps in the existing Airbus in-house pointing control simulation tool box and motion planning algorithms. The aim of this software engineering approach is to develop an enhancing software prototype in several iterative loops. The spiral model divides the development process into four phases. First, the objectives of the development are determined. Second, different realization alternatives are evaluated. Third, the development, implementation and verification are executed and fourth, the next iteration is planned. This approach is useful for developments in which the requirements for the software are not completely determined at the beginning and therefore need to be evolved from iteration to iteration. The four phases of the spiral model are illustrated in Figure 1-7 for the software development done within this thesis work.

8 1. Introduction

1. determine objectives 2. evaluate alternatives

3. develop + verify4. plan next iteration

Requirement identification

Reseach + evaluation of different methods

Implementation and test in reduced simulation environm.

Validation of different methods and performance

comparison

Airbus in-house tools interface requirement identification

Validation of themethods for use ason-board algorithms inlater phases of the project

1. l

oo

p

2. l

oo

p

3. l

oo

p

Evaluation of different implementation

methods

Implementation andtest of selected methods

Figure 1-7: Spiral model for software development

Note that the first two iterations have been completed within this thesis work: In the first iteration, the requirements and constraints for the needed algorithms have been identified, i.e. the previously discussed gaps in the in-house tools and the given constraints for the software development. Different methods to solve these problems have been researched and evaluated. They have been implemented and tested in a reduced simulation environment in Matlab®/Simulink®. Finally, to prepare the next iteration, the different methods have been validated and compared in terms of performance. In the second iteration, the required interfaces to the existing Airbus in-house pointing control design and analysis tools have been identified. Different implementation methods have been evaluated and the one suited best for the integration into the existing tool-chain has been selected, i.e. implementation using embedded Matlab® functions. Finally, the required algorithms have been implemented and tested together with the existing tools.

1.4 Contributions

The following contributions are available as outcome of this thesis work and discussed in detail throughout this report:

• The hexapod open-loop control chain has been analyzed and modelled for simulation in Matlab®/Simulink®:

o The hexapod inverse kinematic has been implemented according to literature resources, relating hexapod states and their first and second order time derivatives to actuator lengths, velocities and accelerations.

1. Introduction 9

o The hexapod forward kinematic has been implemented by combining multiple approaches described in different literature resources, relating actuator lengths, velocities and accelerations to hexapod states and their first and second order time derivatives.

o A simplified model for the linear actuators of the hexapod mechanism has been derived from a more detailed model with support of the Airbus mechanism department.

• The spacecraft attitude dynamic with a hexapod in the loop has been derived and implemented based on Newton-Euler formulation and has been compared to the classical spacecraft rigid-body dynamics without a moving hexapod.

• Different concepts for the pointing system state determination with a hexapod in the loop have been discussed and compared analytically in terms of remaining knowledge errors.

• Line of Sight guidance algorithms for combined spacecraft and hexapod maneuvers have been derived and implemented.

• All software components developed and implemented within this thesis work have been wrapped into a Matlab®/Simulink® ‘Hexapod Simulation Library’ to provide the necessary components for future analyses.

• Different operational scenarios have been developed and compared in terms of feasibility and performance.

• Performance analyses have been done for a representative reference case study similar in parametrization to the Athena spacecraft for different operational scenarios.

• Design trade-offs have been discussed based on the results generated for the reference case study. Design drivers and open design points have been identified based on the above work.

The next chapter provides an overview how these contributions are addressed throughout this report.

1.5 Thesis Outline

This thesis is structured into ten chapters. After the general introduction in Chapter 1, Chapter 2 provides the required theoretical background in classical pointing control for spacecrafts on the one hand and hexapod mechanisms on the other. Thereafter, the pointing control system with a hexapod in the loop is discussed in Chapter 3 and compared to the classical pointing control system without a hexapod. Chapter 4 then derives the models required for simulation of the hexapod open loop control chain. Chapter 5 derives the spacecraft attitude dynamics with a hexapod in the loop. Chapter 6 discusses and compares different state determination approaches with a hexapod in the loop and Chapter 7 derives maneuver guidance algorithms for both, hexapod and spacecraft maneuvers. All developed models and algorithms are then applied to a reference case study in Chapter 8 and design trade-offs based on the simulation results are provided in Chapter 9. Finally, Chapter 10 summarizes the key aspects of the thesis and provides and outlook to potential future research topics related to this field.

11

2 Theoretical Background

Theoretical Background

2.1 Overview

This chapter provides an overview of the theoretical background for this thesis. More details can be found in the provided references. Chapter 2.2 introduces the nomenclature defined for this thesis. Chapter 2.3 discusses a classical pointing control system without a hexapod in the loop. Chapter 2.4 provides a brief introduction into hexapod mechanisms and finally Chapter 2.5 explains the Newton-Euler formulation for multi-body dynamics which is used later to derive the equations of motion of the spacecraft with the moving hexapod in the loop.

2.2 Nomenclature

This chapter introduces the nomenclature for variables used within this thesis. Scalar quantities are always printed in italic font, while vectors or matrices are always printed in bold. If 𝐱 is a vector, then 𝑥 denotes its norm and 𝑥x, 𝑥y, and 𝑥z are its components along the x-, y- and z-axes. A quantity can be

further specified by a leading superscript and several following subscripts. The common scheme for these super- and subscripts is explained hereafter at the following example:

𝐱P(tra)|HH

where the leading superscript H indicates that the quantity 𝐱 is expressed relative to the origin of the {H}-frame. The first subscript P indicates the object to which the quantity belongs. In this case 𝐱P is the state of the hexapod platform. The subscript in brackets further specifies the variable or selects a certain property of it. In this case, the subscript (tra) indicates that only the translational part of the state vector is selected. The last subscript |H indicates in which reference frame the variable is expressed. If a quantity is always related to the same frame, then the corresponding superscript can be left out. Accordingly, if a quantity is always given for the same object or a unique symbol is introduced for the property of a certain object, then this subscript can be left out, too.

2.3 Classical Pointing Control System

The term pointing refers to the task of aligning the LoS of an instrument on board of a SC with the target line, i.e. the line between the SC and the object of interest for observation. In the simplest case, referred to as ‘classical pointing control system’ here, the SC is a rigid body and the instrument is mounted fixed to it. In this case, the SC attitude is directly related to the LoS pointing as illustrated in Figure 2-1 (a). In a second commonly seen case, referred to as ‘advanced classical pointing control system’ here, the instrument is mounted to the SC body via one or more actuated joints. These joints allow for a relative motion between the SC and the instrument LoS. In a space telescope this could for example be a small secondary mirror as discussed in [7]. This complicates the pointing control design because the relative motion between the instrument LoS and the SC needs to be considered for the pointing control design. Thus, the task of aligning the LoS with the target line is divided into SC attitude control and instrument LoS actuator steering control as illustrated in Figure 2-1 (b). However, in this case the mass of the moving part of the instrument is usually much smaller than the mass of the SC platform, such that the effects of those movements on the SC attitude control can usually be neglected.

12 2. Theoretical Background

(a)

XS

ZS=ZLoS

YS

target

(b)

XS

ZS

YS

ZLoS

target

Figure 2-1: (a) Classical and (b) advanced classical pointing control system

The SC attitude is usually directly measured in a feedback loop as illustrated in Figure 2-2. In the classical pointing control system, this also provides a direct measurement of the LoS orientation in inertial space. In an advanced classical pointing control system, the orientation of the moving instrument relative to the SC is required additionally to reconstruct the LoS orientation. In the classical pointing control loop illustrated in Figure 2-2, the attitude error is the input to the pointing controller, i.e. guidance attitude minus measured attitude. Based on this input signal, the controller generates a command input for the attitude actuators. The attitude actuators then generate a torque that acts on the SC together with external disturbances. The SC reacts to these torques according to its dynamic equations. Finally, the resulting SC attitude is measured by sensors and thereby the control loop is closed.

SC guide.

RW model+

-

ϕSC

I SC attitude

ctrl. ++

ϕSC

I

STR

τdist

τRW RW model

Figure 2-2: Classical pointing control loop

More detailed discussion of SC attitude control can be found in [8]–[10]. Typical attitude actuators and sensors are described in Chapter 2.3.1, the SC attitude dynamics formulation is given in Chapter 2.3.2 and external disturbances are described in Chapter 2.3.3.

2.3.1 Classical Pointing Control Components

Two typical examples for SC attitude actuators are reaction wheels and thrusters. As described in [9] and illustrated below, a reaction wheel can be divided into two parts: First, a spinning wheel that can be accelerated and deceleration by a motor torque. Second, a platform assembly that holds the wheel in place. By accelerating the wheel in one direction about the wheel spin-axis, a reaction torque is applied to the platform in the opposite direction due to the conservation of angular momentum.

wheel rotational acceleration

platform rotational acceleration

Figure 2-3: Reaction wheel principle [9]

2. Theoretical Background 13

With three or more reaction wheels mounted with their spin axes not in the same plane, a torque can be created about an arbitrary axis of the SC. More information can be found in [8]–[10]. Markley states in [8] that although reaction wheels are invaluable for providing fine pointing control, they are also one of the major sources of attitude disturbances. For example, small imbalances of the wheel can lead to vibrations that do not only affect the pointing control but are also relevant for other analyses on system level such as micro-vibrations and their transfer through the structure to other components such as sensors and science instruments. Figure 2-4 illustrates the fact that reaction wheels are interdisciplinary error sources that need to be considered in different analyses. Usually different models are used in the different analyses, only representing the effects that are relevant for the analysis at hand. Note that the problem of interdisciplinary error sources may also be relevant for other components, such as a hexapod mechanism, and thus needs to be taken into consideration.

RW µVibrations

RW disturbance torques

Hz

µVibrations

AOCS

Figure 2-4: Interdisciplinary pointing error source reaction wheel

A different type of typical SC attitude actuators are thrusters, which are available in a large variety of dimensions and can thus be used for a variety of maneuvers. One typical use case are agile large angle slew maneuvers. As described in [9], thrusters eject mass of some form to create a force. As illustrated in Figure 2-5 (a), a force vector that does not go through the SC center of mass generates a torque via the lever arm 𝐫. Because thrusters can only provide a force in one direction, two thrusters are needed to allow both, a positive and negative torque about a single axis. Thus, a minimum of six thrusters are needed to produce a torque about an arbitrary axis. More information can be found in [8]–[10].

(a)

thrusterF

r

torque=Fxr (b)

thrusters

torque

Figure 2-5: Torque on a SC due to a single thruster (a) and possible torques for a pair of thrusters (b)

Two typical examples for SC attitude sensors are star trackers and gyros as inertial measurement units (IMU). A STR determines the inertial attitude by locking on and tracking one or several stars. They are usually a two component system with a camera head on the one hand and a separated electronic box for data processing on the other. Due to their high accuracy in absolute three axis attitude measurement, they are the dominating technology today [9]. More information can also be found in [8], [10]. A gyro measures the angular velocity. Often, an attitude estimate is generated by fusioning both, STR and IMU measurements, e.g. with a Kalman filter.


2.3.2 Classical Spacecraft Attitude Dynamics

For classical spacecraft attitude dynamics, the SC can often be considered as a single rigid body and the dynamics are then described by Euler’s rotational equation of motion as derived in [8]:

��I |S = (𝓘(rot)|S)−1(𝐌S|S − [ 𝛚

IS|S]x

𝓘(rot)|S 𝛚I S|S) (2-1)

where ��I |S is the angular acceleration of the SC in inertial frame, 𝓘(rot)|S is the rotational inertia

matrix of the SC and 𝐌S|S is the sum of all torques acting on the SC.

Note that this equation does not consider any flexibilities, e.g. of the solar arrays, or take into account other effects like the gyroscopic terms due to the angular momentum of the reaction wheels. The lather can be neglected here due to the large mass of the SC compared to the mass of the RW spinning wheels. Flexibilities of the structure need to be considered at a later phase of the analysis and are thus not further discussed here.

2.3.3 Classical Disturbances and Model Uncertainties

Typical disturbance torques acting on a spacecraft are solar radiation pressure torque, magnetic torque, fuel sloshing and gravity gradient torque. Additionally, internal disturbance torques for example from the reaction wheels as discussed previously need to be considered for the pointing controller design. More details on internal and external disturbances acting on a SC can be found in [8]. Model uncertainties in classical pointing control systems are usually time-constant and thus handled in the controller design by sufficient stability margins. A typical example for such model uncertainties are inaccuracies in the inertia matrix of the SC.

2.4 Hexapod Mechanism

Throughout this paper, the term hexapod is used to describe a parallel manipulator, which consists of two bodies that are connected to each other by six extensible leg actuators, which can vary in length. Figure 2-6 (a) illustrates such a mechanism, which is also well known as the Stewart platform. This kind of mechanism was first described in the 1960’s by Stewart in [11] as a 6-DoF mechanism for general motion generation for flight simulators. Around twenty years later in the 1980s, the field of parallel manipulators evolved into a more popular research area. Dasgupta provides a good review of the research activities related to the Stewart platform up to 1998 in [12]. Nowadays their application as precise, yet sturdy manipulators has become rather popular in various industries. However, besides a few examples, their application in space is still rather unusual. Two examples for anticipated use of a hexapod on-board a spacecraft are thrust vector control as described in [13] and active vibration isolation as described in [14].

Figure 2-6 illustrates a general 6-6 hexapod geometry with a fixed base and a moving platform connected by six linear actuator legs. The term 6-6 indicates that six base junction points are connected to six platform junction points compared to the 6-3 geometry for the Athena hexapod with only three platform junction points as described in Chapter 1.2.1. In most applications, some form of symmetry can be found in the geometry of base and platform junction points similar to the one shown in Figure 2-6 (b).


(a)

xH

yH

zH

xP yP

zP

h1

h2

h3h4

h5

h6

p1

p2

p3p4

p5

p6

moving platform

fixed base extensible leg actuator

(b)

p1

p2

p3

p4

p5

p6

h1

h2

h3

h4

h5

h6

η1/2

η3/4

η5/6

γH

γP

rH

rP

xH/P

yH/PzH/P

Figure 2-6: General geometry of a 6-DoF Stewart platform in (a) 3D view and (b) top view [15]

2.4.1 Geometry and State Definition

Usually one of the two bodies is seen as the fixed reference. The other body then moves relative to the fixed reference due to the actuator motion. The reference body is called the fixed base or simply base, whereas the other body is called the moving platform, or simply platform. To avoid confusion with the spacecraft body frame, the hexapod base is indicated by the index {H}, for hexapod, whereas the spacecraft body frame is indicated by {B} throughout this report. The moving platform is indicated by index {P}. The six extensible leg actuators couple the moving platform and the fixed base by universal joints. These joints will be simply called junction points from now on. Assuming a symmetric assembly as illustrated in Figure 2-6 (b), the positions of the base junction points 𝐡1…𝐡6 are expressed in and relative to the origin of the {H}-frame as follows:

𝐡H i|H =

[

cos (𝜂i −𝛾H

2) 𝑟H

sin (𝜂i −𝛾H

2) 𝑟H

0

]

with i = 1,3,5 (2-2)

𝐡H i+1|H =

[

cos (𝜂i +𝛾H

2) 𝑟H

sin (𝜂i +𝛾H

2) 𝑟H

0

]

with i = 1,3,5 (2-3)

where 𝑟H is the radius from the center point of the base to the junction points, 𝛾H is the angle between two neighboring points (e.g. 𝐡1 and 𝐡2) and 𝜂𝑖 is the angle between the 𝑥H-axis and the orthogonal onto the line between two neighboring points. The 120° symmetry shown in Figure 2-6 (b) is thus represented by 𝜂1/2 = 90°, 𝜂3/4 = 210° and 𝜂5/6 = 330°.

Correspondingly the platform junction points are defined relative to and expressed in {P}-frame as follows:


𝐩P i|P =

[

cos (𝜂i −𝛾P

2) 𝑟P

sin (𝜂i −𝛾P

2) 𝑟P

0

]

with i = 1,3,5 (2-4)

𝐩P i+1|P =

[

cos (𝜂i +𝛾P

2) 𝑟P

sin (𝜂i +𝛾P

2) 𝑟P

0

]

with i = 1,3,5 (2-5)

Note that the hexapod design for Athena has only three platform junction points (6-3 geometry) as mentioned in Chapter 1.2.1, which can be represented in the above introduced nomenclature by setting 𝛾P = 0.

2.4.2 Position and Orientation

Position and orientation of the platform are described by six states. Three states describe the

translation of the platform, i.e. three Cartesian Coordinates of the platform center of mass 𝐜H P|H

relative to its nominal position 𝐜H P0|H expressed in {H}-frame. The remaining three states describe the

platform orientation, i.e. three Euler angles 𝜙, 𝜃 and 𝜓 – or more precisely: Tait-Bryan angles.

𝐱P|HE = [𝐱P(tra)|H𝐱P(rot)|E

] =

[ 𝑐HP|H,x − 𝑐H P0|H,x

𝑐H P|H,y − 𝑐H P0|H,y

𝑐H P|H,z − 𝑐H P0|H,z

𝜙𝜃𝜓 ]

(2-6)

The first and second-order time derivatives of the above given state vector are:

��P|HE = 𝑑

𝑑𝑡𝐱P|HE =

[��P(tra)|H��P(rot)|E

]

(2-7) = [ ��H P|H,x ��H P|H,y ��H P|H,z �� ]

T

��P|HE = 𝑑2

𝑑𝑡2𝐱P|HE =

[��P(tra)|H��P(rot)|E

]

(2-8) = [ ��H P|H,x ��H P|H,y ��H P|H,z �� ]

T

The three Tait-Bryan angles represent the platform orientation relative to the base plate by three consecutive rotations around x-, y- and z-axes, aligning the {H}-frame axes with the {P}-frame axes. The corresponding rotation matrix transforming a vector expressed in {H}-frame to {P}-frame is given as:

𝐑P H = 𝐑P P0 = 𝑟𝑜𝑡z(𝜓) ⋅ 𝑟𝑜𝑡y(𝜃) ⋅ 𝑟𝑜𝑡𝑥(𝜙)

(2-9)

= [

𝑐(𝜓)𝑐(𝜃) 𝑐(𝜙)𝑠(𝜓) + 𝑐(𝜓)𝑠(𝜙)𝑠(𝜃) 𝑠(𝜙)𝑠(𝜓) − 𝑐(𝜙)𝑐(𝜓)𝑠(𝜃)

−𝑐(𝜃)𝑠(𝜓) 𝑐(𝜙)𝑐(𝜓) − 𝑠(𝜙)𝑠(𝜓)𝑠(𝜃) 𝑐(𝜓)𝑠(𝜙) + 𝑐(𝜙)𝑠(𝜓)𝑠(𝜃)𝑠(𝜃) −𝑐(Θ)𝑠(𝜙) 𝑐(𝜙)𝑐(𝜃)

]


where 𝑐(… ) and 𝑠(… ) are short for cos (… ) and sin (… ). The three rotation matrices 𝑟𝑜𝑡𝑥, 𝑟𝑜𝑡𝑦, 𝑟𝑜𝑡𝑧

are passive rotations, i.e. rotating the coordinate frames not the vector itself.

The inverse of a rotation matrix is equal to its transpose and thus:

𝐑H P = 𝐑P HT (2-10)

As derived in Appendix A.1, the time derivative of the rotation matrix 𝐑H P is computed as:

��H P = [ 𝛚P H|H]xT𝐑H P (2-11)

The second-order time derivative of the rotation matrix is given by:

��H P = [ ��P H|H]xT𝐑H P + [ 𝛚

PH|H]x

T��H P

(2-12)

= ([ ��P H|H]x

T+ ([ 𝛚P H|H]x

T)2

) 𝐑H P

with [ 𝛚P H|H]x being the skew-symmetric matrix of the angular velocity vector 𝛚P H|H:

[𝛚]x =

[

0 −𝜔z 𝜔y𝜔z 0 −𝜔x−𝜔y 𝜔x 0

]

(2-13)

and 𝛚P H|H being the angular velocity of the {H}-frame relative to the {P}-frame expressed in {H}-frame.

Note that this is equivalent to the negative of the angular velocity of the {P}-frame relative to the {H}-frame:

𝛚P H|H = − 𝛚H P|H (2-14)

The angular velocity 𝛚H P|H can be found by mapping the time derivatives of the Tait-Bryan angles of

the consecutive rotations around the x-, y- and z-axes back to the axes of the {H}-frame:

𝛚H P|H = ��𝐞ϕ + �� 𝐑H θ𝐞θ + �� 𝐑H θ 𝐑θ ψ𝐞ψ

(2-15)

= �� [

100] + ��(𝑟𝑜𝑡𝑥(𝜙))

T[010] + �� (𝑟𝑜𝑡y(𝜃) ∗ 𝑟𝑜𝑡𝑥(𝜙))

T[001]

= [

�� + ��𝑠(𝜃)

��𝑐(𝜙) − ��𝑐(𝜃)𝑠(𝜙)

��𝑠(𝜙) + ��𝑐(𝜃)𝑐(𝜙)

]

= [

1 0 𝑠(𝜃)

0 𝑐(𝜙) −c(𝜃)𝑠(𝜙)0 𝑠(𝜙) 𝑐(𝜃)𝑐(𝜙)

] [

��

��

]

= 𝐉A��P(rot)|E

with 𝐉A being the Jacobian matrix.


With that and making use of the fact that [𝛚]xT = −[𝛚]x the first and second-order time derivatives

in Eq. (2-11) and (2-12) can be expressed as:

��H P = [ 𝛚H P|H]x𝐑H P (2-16)

��H P = ([ ��H P|H]x+ ([ 𝛚H P|H]x

)2

) 𝐑H P (2-17)

where the first-order time derivative of 𝛚H P|H, i.e. the angular acceleration, is defined as:

��H P|H = ��A��P(rot)|E + 𝐉A��P(rot)|E

with

��A =

[

0 0 𝑐(𝜃)��

0 −𝑠(𝜙)�� 𝑠(𝜃)��𝑠(𝜙) − 𝑐(𝜃)𝑐(𝜙)��

0 𝑐(𝜙)�� −𝑠(𝜃)��𝑐(𝜙) − 𝑐(𝜃)𝑠(𝜙)��

]

Note that Eq. (2-15) is the angular velocity of the platform expressed in {H}-frame and can also be denoted as the time derivative of the rotational part of the state vector expressed in {H}-frame instead of Tait-Bryan angles:

��P(rot)|H = 𝛚H P|H = 𝐉A��P(rot)|E (2-18)

Thus, the time derivative of the state vector expressed completely in {H}-frame is given by:

��P|H = [��P(tra)|H��P(rot)|H

]

(2-19) = [ ��H P|H,x ��H P|H,y ��H P|H,z ωH P|H,x ωH P|H,y ωH P|H,z]

T

2.5 Newton-Euler Formulation for Multi-Body Dynamics

In literature, usually two methods are suggested for the derivation of the equation of motion for a multibody system, i.e. Newton-Euler formulation and Lagrangian formulation. The former is derived by direct implementation of Newton’s second law of motion. It thus incorporates all forces and torques acting on the individual bodies, including also the coupling forces and torques acting between them. Therefore, further operations are needed to eliminate these virtual forces acting within the system, not performing any work. The Lagrangian formulation is derived in terms of work and energy using generalized coordinates. Thus, workless forces and constraint forces are automatically eliminated, which in general makes this approach simpler and more systematic. It is thus to be preferred for multibody systems with rising complexity. However, for a relatively simple system, the Newton-Euler formulation can be more descriptive and easier to interpret in a physical manner. Therefore, the Newton-Euler method, as described in [16], is briefly introduced for a general multi-body system hereafter.

The motion of a rigid body can be decomposed into a translational motion relative to an arbitrary reference point and a rotational motion around the same point. Correspondingly, the dynamics of a rigid body can be split into a translation of the center of mass, given by Newton’s equation of motion of a mass particle in inertial frame and a rotational motion around the center of mass, given by Euler’s equation of motion. Newton’s equation of motion is defined as:


��|I = ∑𝐅|I (2-20)

Euler’s equation of motion is defined as:

��|I = ∑𝐌|I + ∑(𝐫|I × 𝐅|I) (2-21)

where 𝐐|I and 𝐇|I are the linear and angular momentum of the body in an inertial reference frame.

Their time derivatives are also called kinetic force 𝐅(kin)|I = ��|I and kinetic torque 𝐌(kin)|I = ��|I if

computed in an inertial frame. Though not explicitly indicated by subscript |I for better readability, all quantities in this chapter are therefore expressed in an inertial reference frame. Figure 2-7 shows the free body diagram of a body 𝑖 with center of mass 𝐜i and connection points Oi−1 and Oi+1 to bodies 𝑖 − 1 and 𝑖 + 1 respectively. 𝐅i and 𝐌i are external force and torque applied to the body at the center of mass, 𝐅i−1,i and −𝐅i,i+1 are the coupling forces applied to body 𝑖 by bodies 𝑖 − 1 and 𝑖 + 1

respectively. Similarly, 𝐌i−1,i and −𝐌i,i+1 are the corresponding coupling torques.

For a system of 𝑖 = 1…𝑛 bodies, the coupling forces and torques at the end points, i.e. 𝐅0,1, 𝐌0,1 and

𝐅n,n+1, 𝐌n,n+1, can be seen as environment reaction forces if the end is not a free end. Otherwise they

are zero. The translational equation of motion for body 𝑖 is then obtained as follows:

��i|I = 𝐅i + 𝐅i−1,i − 𝐅i,i+1 (2-22)

The rotational equation of motion for body 𝑖 is obtained as follows:

��i|I = 𝐌i +𝐌i−1,i + 𝐫ci,i−1 × 𝐅i−1,i −𝐌i,i+1 − 𝐫ci,i+1 × 𝐅i,i+1 (2-23)

ci

-Fi,i+1

-Mi,i+1

Oi+1

rc ,i+1irc ,i+1i

Fi-1,i

Mi-1,i

Oi-1

rc ,i-1irc ,i-1i

MiFi

Iωi

Ivi

xI

yI

zI

Figure 2-7: Free body diagram of a rigid body with two connections two other rigid bodies [16]

21

3 Pointing Control System with Hexapod in

the Loop Pointing Control System with Hexapod in the Loop

3.1 Overview

This chapter provides an overview of the pointing control system with the hexapod in the loop. The high-level components introduced here are discussed and analyzed in detail in the following main chapters thereafter. Chapter 3.2 provides a high-level description of the pointing system with the hexapod in the loop and discusses the related repointing process. The differences to the previously introduced classical pointing control system without a hexapod are discussed in Chapter 3.3 thereafter. Finally, related coordinate frames used throughout the rest of this report are defined and illustrated in Chapter 3.4.

3.2 System Description and Repointing Process

The pointing control system with the hexapod in the loop is similar to the advanced classical pointing control system briefly discussed in Chapter 2.3. The major difference however is that the mass of the moving MAM is almost one third of the total SC mass. Thus, every motion of the MAM relative to the SC effects the SC attitude. Due to the high mass of the MAM, these effects cannot be neglected anymore. Figure 3-1 provides a simplified block diagram of the pointing system with the hexapod in the loop. On the bottom left of the diagram, the classical SC attitude control components as discussed in Chapter 2.3 can be found. New or modified components are the hexapod open loop control chain, the SC dynamics taking into account the prescribed hexapod motion and a more complex state determination system. The output of the hexapod open loop control chain are the hexapod states, i.e. position and orientation of the hexapod platform relative to its base. As mentioned before, the hexapod platform is the MAM in this case and the hexapod base is the interface plane between the FMS and the ISM. Adding thermal distortions of the telescope structure to the hexapod states results in the position and orientation of the MAM relative to the SC body frame, which are an input for the SC dynamics with prescribed hexapod motion. Finally, the state determination system provides knowledge about the SC attitude, LoS orientation and the thermal distortions of the telescope. Note that the connection from the state determination block to the hexapod state guidance block is printed with a dotted line, because this feedback is triggered only once before a maneuver and does not lead to a closed-loop control system. The nomenclature used in the figure below is as follows:

𝐑2 1 Rotation matrix from frame 1 to 2

𝐜1 2 Position of frame 2 relative to 1

𝛿 Small change in position or orientation

𝐥 Actuator length vector

… Guidance

… Measurement

22 3. Pointing Control System with Hexapod in the Loop

hex. state guidance

hex. inv. kinematic

actuator model

hex. fwd. kinematic

S/C dynamic with prescr. hex motion

S/C attitude

guidance

S/C attitude

controller

RWmodel

state determination

+-

+-

++++ τdist

τRW

δRthδcth

++

++

RPH

R PH

cP

l l

cP

RBI

RBI

RLP

~

~

RBI

τ

HH

hexapod open-loop

Figure 3-1: Pointing control system with hexapod in the loop: Block diagram

The inputs for the re-pointing procedure between two observations are: On the one hand, the orientation of the new target line in inertial space, instrument selection and focus distance along the LoS between the MAM and detector, which are all uploaded from ground or stored as parameters on-board. On the other hand, a measurement of the thermal distortions of the telescope structure or the resulting error angle between LoS and mirror optical axis, which is measured on-board with the OBM. The re-pointing procedure then includes the following steps: First, the MAM position and orientation is changed through one or several hexapod maneuvers such that the MAM deflects the x-ray beam towards the selected detector, the MOA and Los are aligned and the MAM node is in the correct focus distance from the detector. Second, the SC attitude is changed through a slew maneuver such that the LoS is aligned with the new target line, i.e. the line pointing from the SC towards the new target. After the re-pointing procedure, it must be assured that the LoS remains pointing towards the target with high accuracy in performance and knowledge. This is done with the SC attitude control system only.

3.3 Comparison to Classical Pointing Control System and Related Design Challenges

This chapter compares the previously introduced pointing control system with a hexapod in the loop to the classical pointing control system without a hexapod. Obviously, the hexapod open-loop control chain is a completely new component. The hexapod state guidance needs to be converted to actuator length guidance for the six linear actuators of the hexapod via the inverse kinematic. Effects such as time-delays due to inductivity of the motors, gear friction and backlash due to mechanical tolerances influence the actual execution of the commanded actuator movements. Therefore, these effects need to be modelled and taken into account in the simulations of the pointing system for performance analysis and potentially even on-board for better hexapod state knowledge. The true actuator lengths, i.e. the output of the actuator models, need to be converted back to hexapod states via the forward kinematic. The challenges related to this new component are the implementation of the complex hexapod kinematics that need to be solved in an efficient way for time-simulations and potentially on-

3. Pointing Control System with Hexapod in the Loop 23

board algorithms. Additionally, the hexapod actuators need to be modelled to a reasonable level of detail to be included in the simulations.

The dynamic behavior of the SC is influenced by the moving MAM mass and the SC attitude dynamics cannot be modelled as a single rigid body anymore. Instead, the system consists of two rigid bodies in free space with a relative motion between them. Thus, in general the system has twelve degrees of freedom. Three rotational and three translational degrees of freedom of the whole system in free space and three rotational and three translational degrees of freedom of the hexapod platform relative to the SC. However, for simplicity it is assumed hereafter that the hexapod structure is well dimensioned and rigid enough, such that any flexible modes between the MAM and the ISM/FMS interface can be neglected. This is discussed in more detail in Chapter 5. The challenges related to the different attitude dynamics are to first derive the new dynamics formulation and second, to validate the above stated assumptions for a given hexapod design.

The pointing system state determination is more complex than the one of a pointing system without a hexapod. The STR is mounted on the moving MAM to directly measure its orientation in inertial space at the end of the system chain. If the MOA is aligned with the LoS correctly, then this is also a direct measurement of the LoS orientation. Note that, as mentioned in Chapter 2.3.1, STR are the most accurate attitude measurement units available today. Thus, ideally the LoS orientation and the SC attitude would both be directly measured by two STR units. One mounted on the moving MAM and the other one mounted on the SC. However, due to cost and weight constraints this is not possible. Thus, in general two options exist: First, directly measure the SC attitude in inertial space with one STR and reconstruct the LoS orientation by adding the best knowledge of the MAM orientation relative to the SC. This approach therefore adds the MAM orientation knowledge error to the LoS orientation knowledge error. Second, directly measure the MAM attitude in inertial space with one STR and reconstruct the SC attitude by subtracting the best knowledge of the MAM orientation relative to the SC. This approach therefore adds the MAM orientation knowledge error to the SC attitude knowledge error. Due to the high knowledge requirements on the LoS orientation knowledge, option one has been chosen for Athena. The thermal distortions of the telescope lead to an error angle between the MOA and the LoS. This error angle or the distortions of the telescope itself need to be determined in order to compensate them with corrective maneuvers performed by the hexapod. The challenges related to the pointing system state determination are as follows: First, with the STR mounted on the moving MAM, the SC attitude is not directly measured anymore and needs to be reconstructed from STR measurements and the best knowledge of the relative orientation of the MAM orientation relative to the SC. This adds an additional error source to the SC attitude knowledge compared to the classical pointing system with the STR being mounted body-fixed on the SC. Second, the effects of the thermal distortions need to be measured with some form of on-board metrology and taken into account in the hexapod maneuver guidance for compensation. This leads to relatively complex on-board state determination algorithms as derived in Chapter 6.

Finally, the LoS guidance cannot be directly used as SC attitude guidance anymore. Instead, the LoS is defined by the position of the MAM relative to the SC body frame as described in Chapter 1.2.1. Thus, a complex repointing maneuver guidance is required, involving both, hexapod movements and SC slew. The challenges related to this are that a SC attitude guidance needs to be computed from a given LoS guidance and a given hexapod pose and computationally efficient methods need to be developed for both, hexapod maneuver guidance and SC slew guidance that can be computed on-board. Additionally, an operational flow, meaning chronological order of the hexapod and SC maneuvers, needs to be found that is advantageous for the overall transition time and enables to meet the demanding availability requirements. These topics are discussed in more detail in Chapter 7.


3.4 Reference Coordinate Frames

This chapter defines the coordinate frames used throughout this thesis. Figure 3-2 provides an overview of the different reference frames and the transformations between them. Detailed definitions are provided for each coordinate frame in the subchapters thereafter.

SC att. ref. frame {I}

ECRLoS target

frame

LoS pointing error

Target pointing ref. frame expressed in {I}-frame

Rotation via Sun-Earth angle,translation to SC Position on

Orbit

LoS frame {LoS}Error between nominal and

actual LoS

Nominal LoS frame

Position and orientation of LoS in {B}-frame

SC body frame {B}

SC attitude in {I}-frame

Position and orientation of hexapod base in {B}-frame

Hexapod base ref. frame {H0}

Thermal distortions of telescope structure

Hexapod base frame {H}

Nominal pose of hexapod platform, i.e. MAM

Nominal hex. platf. frame {P0}

MOA to LoS misalignment error

Position and orientation of MAM relative to nominal

pose

Hex. moving platf. frame {P}

Hexapod geometryHex. act. leg ref.

frames {Li}

Hex. local tetrah. frame {Ti} Error Transformation

Coordinate Frame

Transformation

Figure 3-2: Coordinate frames overview and transformations

3.4.1 Earth Centered Rotating Frame {ECR}

The Earth centered rotating frame is used to describe the position and orientation of the spacecraft relative to the Earth rotating around the Sun. It is thus centered with the Earth, its x- and y-axes are in the ecliptic plane with the x-axis aligned with the Sun-Earth line, pointing away from the Sun.

Definition

+xECR Aligned with the Sun-Earth line, pointing away from the Sun. +yECR In the ecliptic plane, orthogonal to Sun-Earth line. +zECR Perpendicular to ecliptic plane, pointing to celestial north; completing the

right-handed orthogonal triad. Origin: Center of the Earth.


3.4.2 Spacecraft Attitude Reference Frame {I}

The spacecraft attitude reference frame provides the nominal observing position with its x-y-plane being orthogonal to the sun-spacecraft line. For the analysis performed within this thesis, it can be considered as an inertial frame and is thus indicated by {I}.

Definition

+xI Parallel with the sun-spacecraft line, pointing away from the sun. +yI Parallel to the ecliptic plane; orthogonal to the sun-spacecraft line and the +xI-

axis. +zI Completing the right-handed orthogonal triad. Origin: Spacecraft center of mass.

Transformation

Translation: Defined by the position of the SC center of mass in the {ECR}-frame:

𝐜ECRI|ECR = [

𝑐ECRI|ECR,x

𝑐ECRI|ECR,y

𝑐ECRI|ECR,z

]

Rotation: 𝐑ECRI = [

cos (𝜎) sin (𝜎) 0−sin (𝜎) cos (𝜎) 00 0 1

] [cos (𝜑) 0 −sin (𝜑)0 1 0

sin (𝜑) 0 cos (𝜑)]

where 𝜎 is the spacecraft-sun-earth angle projected onto the ecliptic and is the spacecraft-sun-earth angle projected onto a plane orthogonal to the ecliptic and containing the sun and Earth.

𝐑IECR = 𝐑ECR

TI

Formula: [

𝑥|ECR𝑦|ECR𝑧|ECR

] = 𝐜ECRI|ECR + 𝐑I

ECR [

𝑥|I𝑦|I𝑧|I]

L2

σ

ϕ

zECR

xECR

yECR

zI

xI

yI

Figure 3-3: Spacecraft attitude reference frame {I} and Earth centered rotating frame {ECR} [17]


3.4.3 Spacecraft Body Frame {B}

The spacecraft body frame is fixed to and rotates with the physical body of the spacecraft. It is thus the principal mechanical reference frame of the spacecraft. Its origin is placed in the center of mass of the spacecraft, which thus rotates around this point. The rotation of {B} relative to {I} is defined as the spacecraft attitude.

Definition

+xB Transverse axis; parallel with launcher interface plane; aligned with xI when SC is pointing to celestial south.

+yB Transverse axis; completing the right-handed orthogonal triad. +zB Longitudinal axis, perpendicular to the launcher interface plane; pointing

towards the focal plane instruments. Origin: Spacecraft center of mass without mirror mass.

Transformation

Translation: None

Rotation: 𝐑IB = 𝑟𝑜𝑡z(𝜓)𝑟𝑜𝑡𝑦(𝜃)𝑟𝑜𝑡x(𝜙)

where 𝜙, 𝜃, 𝜓 are the SC attitude Euler angles. Note that, when the SC is pointing to celestial South, all three Euler angles are zero and the rotation matrix reduces to the identity matrix.

𝐑BI = 𝐑I

TB

Formula: [

𝑥|I𝑦|I𝑧|I] = 𝐑B

I [

𝑥|B𝑦|B𝑧|B]

yB

zB

xB

Figure 3-4: Spacecraft body frame {B} (solar panels folded) [17]


3.4.4 Detector Frames {Di}

The detector frames are used to describe the position and orientation of the three detectors 𝑖=1,2,3 in SC body frame, where the detectors are numbered as follows:

𝑖 = 1 X-IFU detector 𝑖 = 2 WFI large detector 𝑖 = 3 WFI fast detector

Definition

+xDi Transverse axis; completing the right-handed orthogonal triad. +yDi Transverse axis; parallel to the detector plane. +zDi Longitudinal axis, perpendicular to the detector plane; pointing towards the

mirror. Origin: Geometric center of the detector.

Transformation

Translation: Defined by the position of the detector 𝑖 center in {B}-frame:

𝐜B Di|B = [

𝑐B Di|B,x

𝑐B Di|B,y

𝑐B Di|B,z

] = [xi0

12e3 mm]

with 𝑥1 = −450 mm, 𝑥2 = 600 mm and 𝑥3 = 780 mm.

Rotation: 𝐑BDi = 𝑟𝑜𝑡𝑦(𝜋 + 𝜃i)

where 𝜃i is the tilt angle between the LoS when detector 𝑖 is selected and the MAM is perfectly positioned and the SC zB-axis; with 𝜃1 = −2°, 𝜃2 = 3° and 𝜃3 = 3.7°.

𝐑DiB = 𝐑B

TDi

Formula: [

𝑥|B𝑦|B𝑧|B] = 𝐜B Di|B + 𝐑Di

B [

𝑥|Di𝑦|Di𝑧|Di

]

xB

zB

yB

xDi

zDiyDi

Figure 3-5: Detector frame {Di} and SC body frame {B} (solar panels folded)


3.4.5 Line of Sight Frame {LoS}

The detector frame is used to describe the orientation of the LoS in SC body frame. Its origin is placed in the center of the selected detector. Its z-axis provides the LoS orientation.

Definition

+xLoS Transverse axis. +yLoS Transverse axis. +zLoS Longitudinal axis, perpendicular to the detector plane; pointing towards the

mirror. Origin: Geometric center of the detector.

Transformation

Translation: none

Rotation: 𝐑DiLoS = 𝑟𝑜𝑡𝑦(𝜃)𝑟𝑜𝑡𝑦(𝜙)

𝐑LoSDi = 𝐑Di

TLoS

Formula: [

𝑥|Di𝑦|Di𝑧|Di

] = 𝐑LoSDi [

𝑥|LoS𝑦|LoS𝑧|LoS

]

xDi

zDi

yDi/LoS zLoS

xLoS

Figure 3-6: LoS frame {LoS} and detector frame {Di} (solar panels folded)


3.4.6 Hexapod Base Reference Frame {H}

The hexapod base reference frame is fixed to the base of the hexapod and is parallel to the spacecraft body frame. It thus rotates with the spacecraft and can be used to describe the geometry of the hexapod, i.e. positions of the actuator junction points.

Definition

+xH Orthogonal to +yH and +zH; completing the right-handed orthogonal triad. +yH Aligned with +yS of the spacecraft body frame. +zH Aligned with -zS of the spacecraft body frame; pointing towards the center

point of the hexapod moving platform, when in its nominal positon. Origin: Geometric center of the hexagon formed by the junction points of the hexapod

actuator legs with the telescope structure.

Transformation

Translation: Defined by the position of the geometric center of the hexagon formed by the junction points with the telescope structure in {S}-frame:

𝐜B H|B = [

𝑐B H|B,x

𝑐B H|B,y

𝑐B H|B,z

] = [00

−6000mm]

Rotation: 𝐑BH = 𝑟𝑜𝑡𝑦(𝜋) = [

−1 0 00 1 00 0 −1

]

𝐑HB = 𝐑B

TH

Formula: [

𝑥|B𝑦|B𝑧|B] = 𝐜B H|B + [

−1 0 00 1 00 0 −1

] [

𝑥|H𝑦|H𝑧|H]

zH

yH

xH

zH

yH

xH

Figure 3-7: Hexapod base reference frame {H}


3.4.7 Hexapod Platform Reference Frame {P0}

The hexapod platform reference frame provides the nominal, i.e. canonical and upright, position and orientation of the hexapod platform.

Definition

+xP0 Aligned with +xH of the hexapod base reference frame. +yP0 Aligned with +yH of the hexapod base reference frame. +zP0 Aligned with +xH of the hexapod base reference frame. Origin: MAM nodal point/center of mass of the platform in its nominal position.

Transformation

Translation: Defined by the position of the center of mass of the platform in its nominal position in {H}-frame:

𝐜H P0|H = [

𝑐H P0|H,x

𝑐H P0|H,y

𝑐H P0|H,z

] = [00

550mm]

Rotation: 𝐑HP0 = [

1 0 00 1 00 0 1

]

𝐑HS = 𝐑S

TH

Formula: [

𝑥|H𝑦|H𝑧|H] = 𝐜H P0|H + [

𝑥|P0𝑦|P0𝑧|P0

]

zP0

yP0

xP0zH

xH

yH

Figure 3-8: Hexapod platform reference frame {P0} hexapod base reference frame {H}


3.4.8 Hexapod Platform Moving Frame {P}

The hexapod platform moving frame is fixed to and rotates with the platform of the hexapod. Its origin is placed in the center of mass of the platform, which thus rotates around this point. The rotation of {P} relative to {P0} is defined as the platform rotational state and the position of its origin relative to {P0} is defined as the platform translational state.

Definition

+xP Orthogonal to the optical axis of the MAM; on the platform plane. +yP Orthogonal to +xP and +zP; completing the right-handed triad. +zP Aligned with the optical axis of the MAM; positive direction towards target. Origin: MAM nodal point/platform center of mass.

Transformation

Translation: Defined by the position of the center of mass of the platform relative to its nominal position in {P0}-frame:

𝐜P0P|P0 = [

𝐜P0P|P0,x

𝐜P0P|P0,y

𝐜P0P|P0,z

]

Rotation: 𝐑P0P = 𝑟𝑜𝑡z(𝜓)𝑟𝑜𝑡𝑦(𝜃)𝑟𝑜𝑡x(𝜙)

where 𝜙, 𝜃, 𝜓 are the SC attitude Euler angles. Note that, when all three Euler angles are zero, the rotation matrix reduces to the identity matrix.

𝐑PP0 = 𝐑P0

TP

Formula: [

𝑥|P0𝑦|P0𝑧|P0

] = 𝐜P0P|P0 + 𝐑P

P0 [

𝑥|P𝑦|P𝑧|P]

zP0

yP0

xP0

zP

yP

xP

zP0

yP0

xP0

zP

yP

xP

Figure 3-9: Hexapod platform moving frame {P} and hexapod platform reference frame {P0}


3.4.9 Hexapod Actuator Leg Reference Frame {Li}

The hexapod actuator leg reference frame is fixed to the base junction point of a hexapod actuator leg with the telescope structure and is used to describe the actuator leg direction in a local frame.

Definition

+xLi Aligned with the vector from base junction point 𝐡i to 𝐡i+1 for 𝑖 = 1,3,5 or with the vector 𝐡i−1 to 𝐡i for 𝑖 = 2,4,6.

+yLi Orthogonal to +xLi and +zLi; completing the right-handed triad. +zLi Aligned with +zH of the hexapod base reference frame. Origin: Base junction point 𝐡i.

Transformation

Translation: Defined by the position of the base junction point in {H}-frame:

𝐡H i|H = [

ℎH i|H,x

ℎH i|H,y

ℎH i|H,z

]

Rotation: 𝐑HLi = 𝑟𝑜𝑡z(𝜙𝑖)

where 𝜙𝑖 is defined according to the 120° symmetry of the hexapod structure as described in Chapter 2.4:

𝜙1/2 = 𝜂1/2 +𝜋

2= 𝜋

𝜙3/4 = 𝜂3/4 +𝜋

2= 𝜋 +

2

3𝜋

𝜙3/4 = 𝜂5/6 +𝜋

2= 𝜋 +

4

3𝜋

𝐑LiH = 𝐑H

TLi

Formula: [

𝑥|H𝑦|H𝑧|H] = [

ℎH i|H,x

ℎH i|H,y

ℎH i|H,z

] + 𝐑LiH [

𝑥|Li𝑦|Li𝑧|Li]

xH

yH

zH

xLi+1

zLi+1

γHhi+1

hi

pi+1

pi

ηi/i+1

yLi+1

xLi

zLi

yLi

Figure 3-10: Hexapod actuator leg reference frame {Li} and hexapod base reference frame {H}


3.4.10 Hexapod Local Tetrahedron Frame {Ti}

These frames are used in the derivation of the forward kinematic formulation for a 6-3 hexapod geometry in Chapter 4.3.1.1.

Definition

+xTi In the x-y plane of the hexapod base frame, pointing towards base junction point 𝐡i.

+yTi Orthogonal to +xTi and +zTi; completing the right-handed triad. +zTi Aligned with +zH of the hexapod base reference frame. Origin: Hexapod base frame origin.

Transformation

Translation: none

Rotation: 𝐑HTi = 𝑟𝑜𝑡z(𝜙𝑖)

where 𝜙𝑖 is defined according to the 120° symmetry of the hexapod structure as described in Chapter 2.4:

𝜙1 = 𝜂1/2 −𝛾H

2

𝜙2 = 𝜂3/4 −𝛾H

2

𝜙3 = 𝜂5/6 −𝛾H

2

𝐑TiH = 𝐑H

TTi

Formula: [

𝑥|H𝑦|H𝑧|H] = 𝐑Ti

H [

𝑥|Ti𝑦|Ti𝑧|Ti

]

(a) h6

xH

yH

zH

h1

h2

h3h4

h5

p3,4

p1,2p5,6

yT1

xT1

(b) hi

xTi

yTi

zTi

hi+1

pi,i+1

Figure 3-11: Exemplary local tetrahedron frame {T1} in (a) and generic in (b) and hexapod base reference frame {H}

35

4 Hexapod Open Loop Control Chain

Hexapod Open Loop Control Chain

4.1 Overview

In this chapter, the hexapod open loop control chain is discussed in more detail and the required models and algorithms are derived. As briefly discussed in Chapter 2.4, position and orientation of the moving platform define the state of the hexapod. However, to generate actual input commands for the linear actuators, the desired actuator lengths corresponding to a given state of the hexapod are required. Additionally, actuator related effects such as time-delays, backlash and step quantization are modelled in terms of their effect on the actuator length, velocity and acceleration. It is required to analyze how these effects influence the actual motion of the hexapod. In the following, two terms will be used to distinguish between these two domains: ‘Actuator state domain’ and ‘hexapod state domain’. In the hexapod state domain, the hexapod motion is described in terms of the platform position and orientation and their time derivatives, i.e. translational and rotational velocity and acceleration of the platform. In the actuator state domain, the hexapod motion is described in terms of the six linear actuators’ lengths, velocities and accelerations. Both domains are connected by the hexapod kinematic as illustrated in Figure 4-1. Hexapod states can be converted to actuator states via the inverse kinematic. Actuator states can be converted to hexapod states via the forward kinematic.

Hexapod State Domain

Platform position + orientationPlatform transl. + rot. velocity

Platform transl. + rot. acceleration

Actuator State Domain

Actuator lengthsActuator velocities

Actuator accelerations

Forward

Kinematic

Inverse

Kinematic

Figure 4-1: Hexapod state domain and actuator state domain linked by hexapod kinematic

The inverse kinematic problem is rather simple for a hexapod structure and can be solved based on geometry as shown in Chapter 4.2. The forward kinematic is much more complex to solve especially for the general 6-6 geometry (cf. Chapter 2.4). Different approaches exist in literature for specific geometries that simplify the forward kinematic formulation based on geometric properties for a given structure. However, in practice it can be difficult to realize these structures. For example, it can be a challenging task from mechanical engineering perspective to connect two actuators to the platform in the exact same point as required for the 3-6 hexapod geometry suggested in the Athena design in Chapter 1.2.1. Therefore, two approaches to solve the forward kinematic problem have been implemented here, as derived in Chapter 4.3. First, a numerically efficient approach has been implemented that utilizes tetrahedron properties of a perfect 6-3 hexapod geometry. This approach can be used in simulations to generate results with high accuracy with reasonable computation effort. Second, an incremental method has been implemented that relates small changes in actuator lengths to small changes in hexapod states. This approach can be used for on-board computations with lower accuracy requirements, e.g. for a rough estimation of the disturbance torque caused by the moving hexapod platform for a feed forward in the SC attitude controller. Another advantage of this solution is that it is valid for arbitrary 6-6 geometries and thus can take into account any physical realization of the hexapod. Once inverse and forward kinematic are available, actuator effects can be considered in actuator state domain and then be converted back to hexapod state domain to analyze their effects

36 4. Hexapod Open Loop Control Chain

on the SC attitude dynamics as illustrated Figure 4-2. A simplified substitute model is derived in Chapter 4.4, which takes into account the actuator effects that are relevant for the pointing control system.

Hexapod Guidance

(Hexapod State Domain)

Actuator Model

(Actuator State Domain)

SC Dynamic


Inverse

Kinematic

Inverse

Kinematic

Forward

Kinematic

Forward

Kinematic

Hexapod Guidance


Actuator Model

(Actuator State Domain)

SC Dynamic


Inverse

Kinematic

Forward

Kinematic

Figure 4-2: Implementation of actuator model into simulation with hexapod kinematic

4.2 Hexapod Inverse Kinematic

The actuator leg length, velocity and acceleration are defined by the positions, velocities and accelerations of the base and platform junction points. With the base being fixed, the base junction point positions are known and their velocities and accelerations are zero. The positions, velocities and accelerations of the platform junction points can be computed relative to and expressed in base frame {H} as follows, cf. Chapter 2.4:

𝐩H i|H = 𝐜H P|H + 𝐑H P 𝐩P i|P with i = 1…6 (4-1)

��H i|H = ��H P|H + ��H P 𝐩P i|P with i = 1…6 (4-2)

��H i|H = ��H P|H + ��H P 𝐩P i|P with i = 1…6 (4-3)

The leg vectors, i.e. the vectors from a base junction point to the corresponding platform junction point and their time derivatives are obtained as follows:

𝐋i|H = 𝐩H i|H − 𝐡H i|H with i = 1…6 (4-4)

��i|H = ��H i|H with i = 1…6 (4-5)

��i|H = ��H i|H with i = 1…6 (4-6)

The length of each leg can be obtained as the 2-norm of the leg vectors:

𝑙i = ‖𝐋i|H‖2 with i = 1…6 (4-7)

The linear velocity of an actuator along the leg axis, i.e. the velocity of the actuator’s piston, is given by the projection of the time derivative of the leg vector onto the leg axis:

𝑙i = ��i|H ∘ ��i|H with i = 1…6 (4-8)

Finally, the linear acceleration of the actuator is given as:

𝑙i = ��i|H ∘ ��i|H + ��i|H ∘ ��i|H with i = 1…6 (4-9)

where the ∘-operator denotes the scalar product and the unit vector along each leg axis is obtained as:

��i|H = 𝐋i|H

𝑙i with i = 1…6 (4-10)

4. Hexapod Open Loop Control Chain 37

The first-order time derivative of the unit vector along the leg axis can be computed as follows:

��i|H = ��i|H𝑙i−𝐋i|H𝑙i

𝑙i2 with i = 1…6 (4-11)

For implementation purposes and utilizing Matlab’s efficient matrix algebra, the base and platform junction points can be assembled in matrix form:

𝐡H |H = [ 𝐡H 1|H 𝐡H 2|H … 𝐡H 6|H] (4-12)

𝐩H |H = [ 𝐩H 1|H 𝐩H 2|H … 𝐩H 6|H]

(4-13) = [ 𝐜H P|H 𝐜H P|H … 𝐜H P|H] + 𝐑H P[ 𝐩P1|P 𝐩P 2|P … 𝐩P 6|P]

��H |H = [ ��H 1|H ��H 2|H … ��H 6|H]

(4-14) = [ ��H P|H ��H P|H … ��H P|H] + ��H P[ 𝐩P1|P 𝐩P 2|P … 𝐩P 6|P]

��H |H = [ ��H 1|H ��H 2|H … ��H 6|H]

(4-15) = [ ��H P|H ��H P|H … ��H P|H] + ��H P[ 𝐩P1|P 𝐩P 2|P … 𝐩P 6|P]

The matrix of leg vectors and its time derivatives can then be obtained as follows:

𝐋|H = 𝐩H |H − 𝐡H |H (4-16)

��|H = ��H |H (4-17)

��|H = ��H |H (4-18)

The matrix of unit vectors along the leg axes is given by:

��|H = 𝐋|H./ [

𝑙1 … 𝑙6𝑙1 … 𝑙6𝑙1 … . 𝑙6

] (4-19)

��|H =

(��|H.∗ [

𝑙1 … 𝑙6𝑙1 … 𝑙6𝑙1 … . 𝑙6

] − 𝐋|H.∗ [

𝑙1 … 𝑙6𝑙1 … 𝑙6𝑙1 … . 𝑙6

]) ./ [

𝑙12 … 𝑙6

2

𝑙12 … 𝑙6

2

𝑙12 … . 𝑙6

2

]

(4-20)

where the .∗-operator denotes the Hadamard, or element-wise, product and ./ denotes the element-wise quotient.

The leg length, velocity and acceleration vectors are obtained as follows:

𝑙 = [‖𝐋1|H‖2‖𝐋2|H‖2

… ‖𝐋6|H‖2] (4-21)

𝑙 = (diag(��HT ��|H))

T (4-22)


𝑙 = (diag(��H

T ��|H))T+ (diag(��H

T ��|H))T

(4-23)

where diag(… ) returns a column vector containing the diagonal elements of a matrix.

The actuator velocity 𝑙 can also be computed from the state vector time derivative ��P|HE using the

Tait-Bryan angles Jacobian 𝐉E or the kinematic Jacobian 𝐉C as suggested in [18], derived hereafter and used in the hexapod dynamics analysis later on.

𝑙i = ��i|H ∘ ��i|H

(4-24)

= [��i|H

T ��i|HT [ 𝐑H P 𝐩P i|P]x

T] [��P(tra)|H��P(rot)|H

]

Stacking the above given equations for the actuator velocities for all six actuators in matrix form leads to the following form:

𝑙 = 𝐉C��P|H (4-25)

with the kinematic Jacobian being defined as:

𝐉C =

[ ��1|HT [ 𝐑H P 𝐩P 1|P]x

��1|H

��2|HT [ 𝐑H P 𝐩P 2|P]x

��2|H⋮ ⋮

��6|HT [ 𝐑H P 𝐩P 6|P]x

��6|H]

Alternatively using the time derivatives of the Tait-Bryan angles, i.e. ��P(rot)|E, and the Tait-Bryan angles

Jacobian, Eq. (4-25) can be expressed as follows:

𝑙 = 𝐉E��P|HE (4-26)

where

𝐉E = 𝐉C𝐓

with

𝐓 = [𝕀3x3 𝕆3x3𝕆3x3 𝐉A

]

where 𝕀3x3 is the 3x3 identity matrix and 𝕆3x3 denotes the 3x3 matrix with all elements being zero.

The actuator acceleration is then computed as follows:

𝑙 = ��C��P|H + 𝐉C��P|H (4-27)

or written in Tait-Bryan angles:

𝑙 = ��E��P|HE + 𝐉E��P|HE (4-28)

where


��E = ��C𝐓 + 𝐉C��

��C =

[ ��1|HT ([ ��H P 𝐩P 1|P]x

��1|H + [ 𝐑H

P 𝐩P 1|P]x��1|H)

��2|HT ([ ��H P 𝐩P 2|P]x

��2|H + [ 𝐑H

P 𝐩P 2|P]x��2|H)

⋮ ⋮

��6|HT ([ ��H P 𝐩P 6|P]x

��6|H + [ 𝐑H

P 𝐩P 6|P]x��6|H)]

�� = [𝕆3x3 𝕆3x3𝕆3x3 ��A

]

4.3 Hexapod Forward Kinematic

Based on a given actuator configuration, i.e. the length of all six leg actuators, the position and rotation of the platform need to be determined. This problem does not have a unique solution. For the general 6-6 hexapod structure, a 16th order polynomial in one unknown needs to be solved as described in [19]–[21], which leads to 16 potential hexapod poses. Thus, solving the hexapod forward kinematic is a computationally heavy task that includes solving the 16th order polynomial and select the correct solution. Different simplifications of the problem have been derived for special hexapod geometries or by using additional sensors [22], [23]. Here, two different approaches have been chosen and implemented. First, a numerically efficient formulation based on tetrahedron properties of a 6-3 hexapod geometry has been implemented as derived by Song in [24], [25] and described hereafter in Chapter 4.3.1.1. Additionally, an incremental method as proposed by Wang in [26] has been implemented and is described in Chapter 4.3.1.2. This method makes use of the trivial nature of the inverse kinematic derived in Chapter 4.2. A linear relationship between a small change of the joint variables, i.e. actuator length, and the resulting small motion of the platform is derived. The solution to the forward kinematics is then achieved through a series of small changes in the joint variables starting from a known initial pose of the platform. Based on the then known pose of the platform, the rate and acceleration can be computed in closed form as described by Shi in [15] and derived in Chapters 4.3.2 and 4.3.3.

4.3.1 Forward Pose Analysis

4.3.1.1 Tetrahedron Approach

In general, the objective of the forward kinematics is to obtain the positions of the platform junction points for a given set of actuator lengths. Once they are known it is easy to compute the position and orientation of the moving platform. For the 3-6 hexapod geometry only three platform junction points are connected to a pair of base junction points each, as depicted in Figure 4-3 (a). Three line constraints uniquely define a point in three-dimensional space, which can be represented by a tetrahedron. Thus, the 3-6 geometry forms three tetrahedrons defining the platform junction point positions as depicted in Figure 4-3 (b). Two vertices of each tetrahedron represent the two actuator legs that are connected to the platform junction point, i.e. the tip of the tetrahedron. Remembering that their lengths are given inputs for the forward kinematic problem, and with the base junction points being known as well, only one vertex remains unknown to completely define the tetrahedron. The formulation of the forward kinematics in terms of these three unknowns has been described by Song in [24], [25].


(a)

xH

yH

zH

h1

h2

h3h4

h5

h6

p3,4

xP yP

zP

cPxP yP

zP

cP

p1,2p5,6

(b) h6

xH

yH

zH

h1

h2

h3h4

h5

p3,4

p1,2p5,6

Figure 4-3: 6-3 geometry of a 6-DoF Stewart platform in (a) 3D view and (b) illustrating the three tetrahedrons [24]

With no further constraints on the relationship among the platform junction points other than the actuator lengths it seems that the three triangles formed by two base junction points and the corresponding platform junction point (e.g. 𝐡1, 𝐡2, 𝐩1,2) freely rotate about their bottom side.

However, the platform is a rigid body and thus the distance between any two platform junction points

is constant (|𝐩i,i+1 − 𝐩j,j+1| = 𝐶i). Therefore, the three platform junction points must simultaneously

satisfy the following three constraint equations:

𝐶i2 = (𝐩i,i+1|H − 𝐩j,j+1|H) ∘ (𝐩i,i+1|H − 𝐩j,j+1|H) with

i ≠ j

i = j = 1,3,5 (4-29)

= (𝐩i,i+1|H ∘ 𝐩i,i+1|H) + (𝐩j,j+1|H ∘ 𝐩j,j+1|H) −

2(𝐩i,i+1|H ∘ 𝐩j,j+1|H)

Note that Eq. (4-29) must be valid in any reference frame as long as all points are expressed in the same frame. However, it is easier to first derive the positions of the platform junction points in a local frame attached to each of the three tetrahedrons and then convert them into one common frame. The three local tetrahedron frames are denoted with {Ti} here and the {T1}-frame is chosen to be the common frame, which the constraint equation will be expressed in. To simplify the notation in the following derivations, let:

𝛌i|Ti = 𝐩i,i+1|Ti =

[

𝜆i,x𝜆i,y𝜆i,z

]

|Ti

λi = ‖𝐩i,i+1‖

where {Ti} is the local tetrahedron frame as illustrated in Figure 4-4 (a), with the x-axis pointing towards base junction point 𝐡i, z-axis being aligned with the z-axis of the {H}-frame and the y-axis completing

the right-handed orthogonal system. With (𝛌i|T1 ∘ 𝛌i|T1) = 𝜆i2 and (𝛌i|T1 ∘ 𝛌j|T1) = (𝜆i,x𝜆j,x +

𝜆i,y𝜆j,y + 𝜆i,z𝜆j,z)|T1 the constraint Eq. (4-29) ca be rewritten as follows:

𝑓i(𝛌i|T1, 𝛌j|T1) = 𝜆i2 + 𝜆j

2 + (𝜆i,x𝜆j,x + 𝜆i,y𝜆j,y + 𝜆i,z𝜆j,z)|T1− 𝐶i

2 = 0 (4-30)


(a) hi+1

xTi

yTi

zTi

hi

pi,i+1

li

li+1

λi

α

(b)

li

hi

pi,i+1

λi

λi,x xTi

zTi

(c)

p3,4

p1,2

p5,6

120°

120°

120°rP

rP

rP

rP rP

xP

yP

Figure 4-4: Local tetrahedron frame (a), projection of the platform junction point onto the x-axis of the local tetrahedron frame (b) and basis vectors of the platform frame (c)

To express the components of 𝛌i in the local tetrahedron frame, simple triangle relations can be used. To simplify notification hereafter, 𝛌i will always be expressed in its local tetrahedron frame, i.e. 𝛌i =𝛌i|Ti, if not explicitly indicated otherwise. Note also that the origins of the {H}-frame and {Ti}-frames

are all in the same point, i.e. the geometric center of the hexapod base. Thus, point vectors are equivalent, regardless relative to which of these frames they are expressed. The corresponding leading superscripts are thus left out here and if not explicitly indicated otherwise, all points are described relative to the geometric center of the hexapod base, i.e. the origin of {H}- and {Ti}-frames. According to the definition of the {Ti}-frame, the component 𝜆i,x along the x-axis is equivalent to the projection

of 𝛌i onto vector 𝐡2i−1|Ti, as illustrated in Figure 4-4 (b):

𝜆i,x = 𝛌i∘𝐡2i−1|Ti

ℎ2i−1 = 𝜆icos (𝛼)

(4-31)

where cos (𝛼) can be determined from the law of cosines:

cos (𝛼) = 𝑙2i−12 −𝜆i

2−ℎ2i−12

−2𝜆iℎ2i−1

(4-32)

Inserting Eq. (4-32) into (4-31) leads to:

𝜆i,x = 𝜆i2+ℎ2i−1

2 −𝑙2i−12

2ℎ2i−1

(4-33)

The component 𝜆i,y along the y-axis of the {Ti}-frame can be found from the projection of 𝛌i onto

vector 𝐡2i|Ti, which is in the x-y-plane. Thus from:

𝛌i ∘ 𝐡2i|Ti = 𝜆i,xℎ2i,x|Ti + 𝜆i,yℎ2i,y|Ti + 0𝜆i,z (4-34)

where 𝜆i,x is given in Eq. (4-33) and the projection of 𝛌i onto 𝐡2i can be determined similar to before:

𝛌i ∘ 𝐡2i|Ti = 𝜆i2+ℎ2i

2 −𝑙2i2

2 (4-35)

Inserting Eq. (4-35) into (4-34) and solving for 𝜆i,y leads to:

𝜆i,y = 𝜆i2+ℎ2i

2 −𝑙2i2 −2𝜆i,xℎ2i,x|Ti

2ℎ2i,y|Ti

(4-36)


where the x- and y-components of 𝐡2i|Ti in {Ti} frame are computed as follows:

ℎ2i,x|Ti = 𝐡2i|H ∘ 𝐱Ti|H

ℎ2i,y|Ti = 𝐡2i|H ∘ 𝐲Ti|H

where 𝐱Ti|H and 𝐲Ti|H are the unit vectors along the x- and y-axes of the {Ti}-frame expressed in {H}-

frame. Finally, the component 𝜆i,z along the z-axis of the {Ti}-frame can be computed as follows:

𝜆i,z = √𝜆i

2 − 𝜆i,x2 − 𝜆i,y

2 (4-37)

Thus, for given actuator lengths 𝑙1… 𝑙6 the platform junction points 𝐩i,i+1|Ti = 𝛌i can be expressed in

terms of the three unknown vector lengths 𝜆i. However, the three constraint Eq. (4-30) are non-linear and implicit and can thus be solved numerically, e.g. using three-dimensional Newton-Raphson method. Once the solutions for 𝜆1, 𝜆2, 𝜆3 have been computed within a prescribed convergence tolerance, the platform junction points 𝛌1, 𝛌2, 𝛌3 can be expressed in the local tetrahedron frames from Eq. (4-33), (4-35) and (4-37). Those can then be converted into the hexapod base frame and finally the position of the platform relative to and expressed in {H}-frame can be determined from the following geometrical relation:

𝐜H P|H = 1

3∑ 𝐩i,i+1|Hi =

1

3∑ 𝛌i|Hi (4-38)

For the given 120° symmetry of the platform junction points as depicted in Figure 4-4 (c), the basis vectors of the {P}-frame expressed in {H}-frame are computed as follows:

𝐱P|H = 𝐩5,6|H−𝐩3,4|H

√3 𝑟P

(4-39)

𝐲P|H = 𝐩1,2|H− 𝐜H P|H

𝑟P

(4-40)

𝐳P|H = 𝐱P|H × 𝐲P|H (4-41)

The rotation matrix from {H}-frame to {P}-frame is then defined as follows. From there it is easy to compute the three Euler angles used to describe the orientation of the platform:

𝐑𝐏 𝐇 = [𝐱P|H 𝐲P|H 𝐳P|H] (4-42)

Note that the numerical algorithm that solves for the three unknown needs to include the following case-sensitive behavior to account for the signs of the following dot-products:

𝛌i ∘ 𝐡2i−1|Ti = {+𝜆i ℎ2i−1 if 𝜆i

2 + ℎ2i−12 ≥ 𝑙2i−1

2

−𝜆i ℎ2i−1 otherwise

(4-43)

𝛌i ∘ 𝐡2i|Ti = {+𝜆i,xℎ2i,x + 𝜆i,yℎ2i,y if 𝜆i

2 + ℎ2i2 ≥ 𝑙2i

2

−𝜆i,xℎ2i,x − 𝜆i,yℎ2i,y otherwise

(4-44)

Thus, the following algorithms need to be implemented for the computation of the elements of 𝛌i:


𝜆i,x =

{

𝜆i2+ℎ2i−1

2 −𝑙2i−12

2ℎ2i−1if 𝜆i

2 + ℎ2i−12 ≥ 𝑙2i−1

2

−𝜆i2+ℎ2i−1

2 −𝑙2i−12

2ℎ2i−1otherwise

(4-45)

𝜆i,y =

{

𝜆i2+ℎ2i

2 −𝑙2i2 −2𝜆i,x(𝐡2i|H∘𝐱Ti|H)

2(𝐡2i|H∘𝐲Ti|H)if 𝜆i

2 + ℎ2i2 ≥ 𝑙2i

2

−(𝜆i2+ℎ2i

2 −𝑙2i2 )−2𝜆i,x(𝐡2i|H∘𝐱Ti|H)

2(𝐡2i|H∘𝐲Ti|H)otherwise

(4-46)

𝜆i,z =

{

√𝜆i2 − 𝜆i,x

2 − 𝜆i,y2 if 𝜆i

2 ≥ 𝜆i,x2 + 𝜆i,y

2

|√𝜆i2 − 𝜆i,x

2 − 𝜆i,y2 | otherwise

(4-47)

As mentioned before, for constraint Eq. (4-30) to hold, the vectors 𝛌1, 𝛌2, 𝛌3 need to be converted into a common reference frame, which was chosen to be {T1} for convenience here, but could also be any other frame. Therefore, starting from arbitrarily chosen initial values for the three unknown scalars λ1, λ2, λ3 within the feasible region, 0 < λi < 2𝑙i, or if possible from the values corresponding to the last known pose of the hexapod, the vector components of 𝛌1, 𝛌2, 𝛌3 can be computed according to Eq. (4-45), (4-46) and (4-47) in the local tetrahedron frames. Once converted to the common reference frame {T1}, the components and lengths of 𝛌1|T1, 𝛌2|T1, 𝛌3|T1 can be plugged into the three

simultaneous constraint functions in order to compute the function values:

F(𝛌1|T1, 𝛌2|T1, 𝛌3|T1) =

[

𝑓1(𝛌1|T1, 𝛌2|T1)

𝑓2(𝛌2|T1, 𝛌3|T1)

𝑓3(𝛌3|T1, 𝛌1|T1)

]

(4-48)

Next, new values for the three unknowns λ1, λ2, λ3 are computed according to the three-dimensional Newton-Raphson method until the constraint function values in Eq. (4-48) converge to zero within a prescribed convergence tolerance. The increments/decrements ∆λ1, ∆λ2, ∆λ3 are computed in each iteration as follows, using the current constraint function values and the Jacobian matrix of the constraint function:

∆λ = [

∆λ1∆λ2∆λ3

] = 𝐉F

−1F(𝛌1|T1, 𝛌2|T1, 𝛌3|T1)

(4-49)

The new values for λ1, λ2, λ3 are then computed as follows:

λi,n = λi,n−1 − ∆λi (4-50)

where n denotes the number of the current iteration of the algorithm. The Jacobian 𝐉F is defined as follows:

𝐉F =

[ 𝜕𝑓1

𝜕λ1

𝜕𝑓1

𝜕λ2

𝜕𝑓1

𝜕λ3𝜕𝑓2

𝜕λ1

𝜕𝑓2

𝜕λ2

𝜕𝑓2

𝜕λ3𝜕𝑓3

𝜕λ1

𝜕𝑓3

𝜕λ2

𝜕𝑓3

𝜕λ3]

|T1

(4-51)

where the partial derivatives of the of the constraint functions expressed in {T1}-frame are defined as:


∂𝑓i(𝛌i|T1,𝛌j|T1)

∂λk = 2λi

𝜕λi

𝜕λk+ 2λj

𝜕λj

𝜕λk− 2∑ (

𝜕λi,a|T1

𝜕λkλj,a|T1 + λi,a|T1

𝜕λj,a|T1

𝜕λk)𝑎=𝑥,𝑦,𝑧

(4-52)

where

𝜕λi

𝜕λk = {

1 if i = k0 otherwise

(4-53)

The partial derivatives of the components of 𝛌1|T1, 𝛌2|T1, 𝛌3|T1 are computed as follows:

[ 𝜕λi,x|T1

𝜕λk𝜕λi,y|T1

𝜕λk𝜕λi,z|T1

𝜕λk ]

= 𝐑T1Ti

[ 𝜕λi,x|Ti

𝜕λk𝜕λi,y|Ti

𝜕λk𝜕λi,z|Ti

𝜕λk ]

(4-54)

with

𝜕λi,x|Ti

𝜕λk =

{{2𝜆i(2ℎ2i−1)

−1 if 𝜆i2 + ℎ2i−1

2 ≥ 𝑙2i−12

−2𝜆i(2ℎ2i−1)−1 otherwise

} if i = k

0 otherwise

(4-55)

𝜕λi,y|Ti

𝜕λk =

{

{(2𝜆i − 2ℎ2i,x|Ti

𝜕λi,x|Ti

𝜕λk) (2ℎ2i,y|Ti)

−1if 𝜆i

2 + ℎ2i2 ≥ 𝑙2i

2

−(2𝜆i − 2ℎ2i,x|Ti𝜕λi,x|Ti

𝜕λk) (2ℎ2i,y|Ti)

−1otherwise

} if i = k

−2ℎ2i,x|Ti𝜕λi,x|Ti

𝜕λk(2ℎ2i,y|Ti)

−1otherwise

(4-56)

𝜕λi,y|Ti

𝜕λk =

{

1

2(𝜆i2 − 𝜆i,x

2 − 𝜆i,y2 )

−1

2 (2𝜆𝑖𝜕λi

𝜕λk− 2λi,x|Ti

𝜕λi,x|Ti

𝜕λk− 2λi,y|Ti

𝜕λi,y|Ti

𝜕λk) if i = k

|1

2(𝜆i2 − 𝜆i,x

2 − 𝜆i,y2 )

−1

2 (2𝜆𝑖𝜕λi

𝜕λk− 2λi,x|Ti

𝜕λi,x|Ti

𝜕λk− 2λi,y|Ti

𝜕λi,y|Ti

𝜕λk)| otherwise

(4-57)

4.3.1.2 Incremental Approach

This chapter provides an alternative approach that differs from the previously described algorithm in Chapter 4.3.1.1 in two ways. First, it is derived for a general 6-6 hexapod geometry. Second, it makes use of linear approximations and can thus only be used for small changes or computations with low accuracy requirements. For simplification purposes of the following derivations, the terminology and concepts from the theory of Lie groups will be used and are thus introduced briefly. For more details see [27]. The pose of the hexapod platform can be fully described by a translation along a vector and a rotation around its center point and is thus an element of the Euclidean motion group, SE(3). The Euclidean motion group is a semi direct product of ℝ3 with the special orthogonal group (or ‘three-dimensional rotation group’), SO(3). Elements of SE(3) are denoted as 𝑔 = (𝐚, 𝐀) ∈ SE(3) here, where 𝐀 ∈ SO(3) is a rotation matrix and 𝐚 ∈ ℝ3 a three-dimensional vector.

Any element of SE(3) can be represented as a 4x4 homogenous transformation matrix:

𝑔(𝐚, 𝐀) = [𝐀 𝐚𝕆1x3 1

] (4-58)

Note that the homogeneous transformation matrix includes both, a translation 𝐚 and a rotation 𝐀. Pre-

multiplying the homogeneous representation 𝐱 = [𝐱𝐓 1]𝐓 of a vector 𝐱 ∈ ℝ3 with the homogeneous


transformation matrix yields the homogeneous representation 𝐲 of the vector 𝐲 ∈ ℝ3, which can

otherwise be obtained by separately applying translation and rotation to 𝐱, i.e. 𝐲 = 𝑔𝐱 = 𝐀𝐱 + 𝐚:

𝐲 = [𝐲1] =

[𝐀 𝐚𝕆1x3 1

] [𝐱1]

= [𝐀𝐱 + 𝐚1

] (4-59)

The group law for elements of SE(3) is defined as follows:

𝑔1 ⋅ 𝑔2 = (𝐚𝟏 + 𝐀𝟏𝐚𝟐, 𝐀𝟏𝐀𝟐) (4-60)

and

𝑔−1 = (−𝐀𝑇𝐚, 𝐀𝑇) (4-61)

Any element of SE(3) can be written as the product of a pure translation and pure rotation:

(𝐚, 𝐀) = (𝐚, 𝕀3𝑥3) ⋅ (𝕆1x3, 𝐀) (4-62)

Following this terminology, the actuator leg length can be expressed as follows:

𝑙i = ‖𝑔 ⋅ 𝐩P i|P − 𝐡H i|H‖2 (4-63)

where 𝐩P i|P and 𝐡H i|H are the platform and base junction points relative to and expressed in the

corresponding coordinate frames and 𝑔( 𝐜H P|H, 𝐑H

P) ∈ SE(3) is the transformation from {P} to {H}-

frame.

The equation above is another way to express the inverse kinematics derived in Chapter 4.1 and once again illustrates their simplicity. In contrast to that, obtaining the forward kinematics of the form 𝑔 =𝑔(𝑙1, 𝑙2, … , 𝑙6), i.e. deriving the platform pose from a given leg configuration, is much harder, as discussed before. It is thus solved here using the incremental method by Wang in [26]. This approach utilizes the trivial nature of the inverse kinematics by making small steps in actuator length towards the required actuator configuration starting from the last known pose of the platform.

A small change 𝜖𝑖 in leg length leads to a small rotation 𝛀 and a small translation 𝐯 of the platform:

𝑙i + 𝜖𝑖 = ‖𝑔 ⋅ 𝛾 ⋅ 𝐩P i|P − 𝐡H i|H‖2 (4-64)

where

𝛾 = [𝕀3𝑥3 + 𝛀 𝐚𝕆1x3 1

]

is a small motion in 𝑆𝐸(3) such that ‖𝛏‖2 < 1, where

𝛏 = [𝛾 − 𝕀4𝑥4]V =

[ 𝛚H

P|H

𝐯 ]

The V–operator converts a 4x4 screw matrix into a 6x1 vector. The 𝛏-vector thus contains the angular

increment 𝛚H P|H and translational increment 𝐯 of the platform pose due to the small leg length

increment 𝜖𝑖.

Subtracting Eq. (4-63) from (4-64) relates incremental changes in leg length to incremental changes in the platform pose:


𝜖i = ‖𝑔 ⋅ 𝛾 ⋅ 𝐩P i|P − 𝐡H i|H‖2− ‖𝑔 ⋅ 𝐩P i|P − 𝐡H i|H‖2

(4-65) = ‖𝛾 ⋅ 𝐩P i|P − 𝑔

−1 ⋅ 𝐡H i|H‖2− ‖ 𝐩P i|P − 𝑔

−1 ⋅ 𝐡H i|H‖2

Note that in the first line of Eq. (4-65) all vectors are relative to and expressed in (or converted to) {H}-frame. In the second line they are converted to {P}-frame by a multiplication with 𝑔−1. This operation does not change the norm of the vectors and thus both lines are equivalent.

Expanding 𝛾 𝐩P i|P = 𝐩P i|P +𝛀 𝐩P i|P + 𝐯 in Eq. (4-65) leads to the following expression:

𝜖i = ‖ 𝐩P i|P − 𝑔−1 𝐡H i|H + (𝛀 𝐩P i|P + 𝐯)‖2

− ‖ 𝐩P i|P − 𝑔−1 𝐡H i|H‖2

(4-66)

Because 𝛾 is a small motion, (𝛀 𝐩P i|P + 𝐯) is a small motion and the first-order Taylor series

approximation, with 𝛅 being small as derived in [26]:

‖𝐛 + 𝛅‖ ≈ ‖𝐛‖ +𝐛T𝛅

‖𝐛‖

can be applied to the left hand norm in Eq. (4-66):

𝜖i ≈ ‖ 𝐩P i|P − 𝑔

−1 𝐡H i|H‖2+( 𝐩P i|P−𝑔

−1 𝐡H i|H)T(𝛀 𝐩P i|P+𝐯)

‖ 𝐩P i|P−𝑔−1 𝐡H i|H‖2

…

−‖ 𝐩P i|P − 𝑔−1 𝐡H i|H‖2

(4-67)

= ( 𝐩P i|P−𝑔−1 𝐡H i|H)

T


(𝛀 𝐩P i|P + 𝐯)

With 𝛀 = [ 𝛚H P|H]x , [𝐚]𝑥 𝐛 = [𝐛]𝑥

𝑇𝐚 and ‖𝐚‖ = ‖−𝐚‖ Eq. (4-67) becomes:

𝜖i ≈ ( 𝐩P i|P−𝑔−1 𝐡H i|H)

T


(−[ 𝐩P i|P]x𝛚H P|H + 𝐯)

(4-68)

= (𝑔−1 𝐡H i|H− 𝐩P i|P)T


([ 𝐩P i|P]x𝛚H P|H − 𝐯)

= (𝑔−1 𝐡H i|H− 𝐩P i|P)T

‖𝑔−1 𝐡H i|H− 𝐩P i|P‖2[[ 𝐩P i|P]x

−𝕀3𝑥3] [𝛚H P|H

𝐯]

= 𝐦i𝛏

where

𝐦i =

(𝑔−1 𝐡H i|H− 𝐩P i|P)T

‖𝑔−1 𝐡H i|H− 𝐩P i|P‖2[[ 𝐩P i|P]x

−𝕀3𝑥3]

Stacking these equations for 𝑖 = 1…6 leads to the following linear system of equations:

𝛜 = 𝐌𝛏 (4-69)

where 𝛜 = [𝛜𝟏 … 𝛜𝟔]𝐓 and 𝐌 = [𝐦𝟏𝐓 … 𝐦𝟔

𝐓]𝐓.


Eq. (4-69) can be inverted to solve for 𝛏 and thus find the angular and position increment of the platform pose due to the leg length increments. By making 𝛜 sufficiently small, the forward kinematics problem can be solved iteratively. Note that 𝛏 includes the angular increment of the platform pose in terms of Euler axis/angle representation, not Euler angles. However, if the step size is small enough, the difference between three subsequent rotations by the Euler angles and one instant rotation around the Euler axis can be neglected. Otherwise the Euler axis/angle representation needs to be converted into Euler angles of the corresponding rotation sequence.

4.3.1.3 Comparison

In order to compare the two approaches introduced in Chapters 4.3.1.1 and 4.3.1.2, a distance metric is used, which quantifies how close the pose generated from the forward kinematic algorithm is to the actual pose of the platform. The forward kinematic algorithms have been run for different step sizes (incremental approach) or different convergence criteria for the Newton-Raphson solver algorithm (tetrahedron approach) respectively. A Monte-Carlo simulation has been run in an outer loop starting from the same initial pose of the hexapod but computing the forward kinematics for different end poses. The mean has then been computed over the results of the Monte-Carlo runs. Figure 4-5 shows the computation time over the achieved accuracy of the solution measured by the distance metric for SE(3) as given by [27]:

D(𝑔1, 𝑔2) = √‖𝐚1 − 𝐚2‖

2 + 𝛼‖log𝐀1T 𝐀2‖

2

(4-70)

where 𝑔1 = 𝑔1(𝐚1, 𝐀1) and 𝑔2 = 𝑔2(𝐚2, 𝐀2) are elements of SE(3) , ‖∙‖ is the Euclidean norm and 𝛼 is a parameter, which can be used to balance between position and orientation error. According to [26] this parameter needs to be chosen such that position and orientation error are within the same order of magnitude. Note that the absolute computation time in Figure 4-5 is not of much interest, as it highly depends on the system the computations are run on. Of much hinger interest are the qualitative characteristics of the curves and their relation to each other. As can be seen, the iterative approach by Wang is very efficient for relatively low accuracy requirements and could thus be used on board for a rough estimate of the hexapod kinematics. This could for example be used for a feed forward of the disturbance torque induced by the hexapod motion. However, with decreasing step size and thus increasing accuracy (i.e. smaller error metric), the computation time increases exponentially. Thus, the incremental approach is not very well suited for ground simulations and analyses with high accuracy requirements. In this case, the tetrahedron approach of Song is superior and is thus used for the analyses and simulations performed throughout this thesis.

Figure 4-5: Forward pose analysis computation time over error metric for different approaches


4.3.2 Forward Rate Analysis

Given the actuator velocities and the pose of the hexapod, i.e. position and orientation found from one of the two methods derived in the previous chapter, the platform translational and angular

velocities, ��H P|H and 𝛚P H|H, can be obtained by solving a linear system of equations as derived in [15]

and discussed hereafter. For the derivation of the linear system of equations two auxiliary variables will be introduced for each leg actuator. The platform translational and angular velocities are then expressed in terms of these auxiliary variables and additional boundary conditions are established to eliminate these unknowns.

The auxiliary variables are 𝛼i and 𝛽i, with 𝛽i being the angle between leg vector 𝐋i and the zLi-axis of the Local Actuator Leg frame {Li} and 𝛼i being the angle between the projection of 𝐋i onto the base plane (xLi-yLi–plane; xH-yH–plane;) and the xLi-axis as depicted in Figure 4-6.

The leg vector can be expressed in {Li}-frame as a function of these auxiliary variables as follows:

𝐋i|L =

[

𝑙i sin(𝛽i) cos (𝛼i)

𝑙i sin(𝛽i) sin (𝛼i)

𝑙i cos(𝛽i)]

(4-71)

For a known leg vector and length, the auxiliary variables can be determined as follows:

𝛼i = atan2(��i|L,y, ��i|L,x) (4-72)

𝛽i = acos(��i|L,z) (4-73)

where ��i|L = 𝐑Li H��i|H and atan2(y, x) returns the angle between the positive x-axis and the point

(x, y).

The platform junction points can be expressed in {H}-frame in terms of 𝑙i, 𝛼i and 𝛽i as follows:

𝐩H i|H = 𝐡H i|H + 𝐑H Li𝐋i|L

(4-74)

= 𝐡H i|H + [

𝑙i 𝑠(𝛽i) 𝑐 (𝛼i +Φi)

𝑙i 𝑠(𝛽i) 𝑠 (𝛼i +Φi)

𝑙i 𝑐(𝛽i)]

where Φ𝑖 is the angle between +xH and +xLi-axes as defined in Chapter 3.4.9.

The velocities of the platform junction points are then:

��H i|H =

[

𝑙i 𝑠(𝛽i) 𝑐 (𝛼i +Φi)

𝑙i 𝑠(𝛽i) 𝑠 (𝛼i +Φi)

𝑙i 𝑐(𝛽i)

] + [

𝑙i��i 𝑐(𝛽i) 𝑐(𝛼i +Φi) − 𝑙i��i 𝑠(𝛽i) 𝑠 (𝛼i +Φi)

𝑙i��i 𝑐(𝛽i) 𝑠(𝛼i +Φi) + 𝑙i��i 𝑠(𝛽i) 𝑐 (𝛼i +Φi)

−𝑙i��i 𝑠(𝛽i)

]

(4-75)

Finally, the platform translational velocity ��H P|H can be expressed in terms of the auxiliary variables

by applying the centroidal property:

��H P|H = 1

3∑ ��H i|H𝑖 (4-76)

where three platform junction points need to be chosen, which form a triangle centered in 𝐜P.

Choosing 𝑖 = 2,4,6 here, ��H P|H is expressed in terms of the six auxiliary variables ��2, ��4, ��6 and

��2, ��4, ��6. Note that 𝛼2, 𝛼4, 𝛼6 and 𝛽2, 𝛽4, 𝛽6 can be computed according to Eq. (4-72) and (4-73) from the known pose of the platform.


In the next step, the platform angular velocity 𝛚H P|H will be expressed in terms of the auxiliary

variables, too. Therefore, the equation for the platform junction point velocities derived in Chapter 4.2 is reused:

��H i|H = ��H P|H + ��H P 𝐩P i|P

(4-77)

= ��H P|H + [ 𝛚H

P|H]x𝐑H P 𝐩P i|P

= ��H P|H + [ 𝛚H

P|H]x𝐩P i|H

Rearranging Eq. (4-77) in matrix form for 𝑖 = 2,4,6 leads to:

𝐂 = [ 𝛚H P|H]x𝐏 (4-78)

where 𝐂 = [ ��H 2|H ��H 4|H ��H 6|H] − [ ��HP|H ��H P|H ��H P|H] and 𝐏 = [ 𝐩P 2|H 𝐩P 4|H 𝐩P 6|H].

By inserting Eq. (4-75) and (4-76) into this equation, the platform angular velocity can be expressed in terms of the auxiliary variables, too. Unfortunately, due to the fact that all platform junction points are coplanar, 𝐏 is singular and can thus not be inverted to solve for the platform angular velocity. However,

[ 𝛚H P|H]x is skew-symmetric and therefore it is possible to select any three independent scalar

equations from Eq. (4-78) to solve for the three elements of 𝛚H P|H. Which ones of the nine scalar

equations are independent depends on the current pose of the platform. Thus, for a computer based algorithm that solves the forward rate analysis problem, all nine of them should be implemented as a linear equation system of rank three:

𝛚H P|H = 𝐀(rate)−1 𝐛(rate) (4-79)

With Eq. (4-76) and (4-79), the forward rate analysis problem is expressed in terms of the six auxiliary

variables ��2, ��4, ��6 and ��2, ��4, ��6. To eliminate these auxiliary variables, six constraint equations are required. The first three scalar constraint equations can be found by utilizing the kinematic constraint of rigid body motion illustrated in Figure 4-7, i.e. the fact that the projections of two points’ velocities onto the straight line through these two points must be equal:

��H i|H ∘ ( 𝐩H i|H − 𝐩P j|H) = ��H j|H ∘ ( 𝐩H i|H − 𝐩P j|H) (4-80)

with 𝑖 = 2,4,6, 𝑗 = 2,4,6, 𝑖 ≠ 𝑗

The remaining three scalar equations can be found by equating the velocities of the other three

actuators, i.e. 𝑙i for 𝑖 = 1,3,5, with the projection of the corresponding platform junction point velocities onto the leg axes:

𝑙i = ��H i|H ∘ ��i|L (4-81)

where ��H i|H for 𝑖 = 1,3,5 can be expressed in terms of the six auxiliary variables ��2, ��4, ��6 and

��2, ��4, ��6 by inserting Eq. (4-76) and (4-79) into (4-77).


4.3.3 Forward Acceleration Analysis

Given the actuator accelerations and the platform position and orientation as well as the corresponding velocities from the previous two chapters, the platform translational and angular

accelerations, ��H P|H and ��P H|H, can be obtained by solving a linear system of equations as derived in

[15] and discussed hereafter.

The accelerations of the platform junction points in terms of the auxiliary variables are given by the time derivative of Eq. (4-75):

��H i|H =

[

𝑙i 𝑠(𝛽i) c(𝛼i +Φi) + 𝑙i��i 𝑐(𝛽i) 𝑐(𝛼i +Φi) − 𝑙i��i 𝑠(𝛽i) 𝑠 (𝛼i +Φi)

𝑙i 𝑠(𝛽i) s(𝛼i +Φi) + 𝑙i��i c(𝛽i) 𝑠(𝛼i +Φi) + 𝑙i��i 𝑠(𝛽i) 𝑐(𝛼i +Φi)

𝑙i 𝑐(𝛽i) − 𝑙i��i 𝑠(𝛽i)

] …

+[

𝑙i��i 𝑐(𝛽i) 𝑐(𝛼i +Φi) + 𝑙i��i 𝑐(𝛽i) 𝑐(𝛼i +Φi)

𝑙i��i 𝑐(𝛽i) 𝑠(𝛼i +Φi) + 𝑙i��ii 𝑐(𝛽i) 𝑠(𝛼i +Φi)

−𝑙i��i 𝑠(𝛽i) − 𝑙i��ii 𝑠(𝛽i)

…

−𝑙i��i2 𝑠(𝛽i) 𝑠(𝛼i +Φi) − 𝑙i��i��i 𝑐(𝛽i) 𝑐(𝛼i +Φi)

−𝑙i��i2 𝑠(𝛽i) 𝑠(𝛼i +Φi) + 𝑙i��i��i 𝑐(𝛽i) 𝑐(𝛼i +Φi)

−𝑙i��i2 𝑐(𝛽i)

]…

+[−𝑙i��i 𝑠(𝛽i) 𝑠(𝛼i +Φi) − 𝑙i��i 𝑠(𝛽i) 𝑠 (𝛼i +Φi)

𝑙i��i 𝑠(𝛽i) 𝑐(𝛼i +Φi) + 𝑙i��i 𝑠(𝛽i) 𝑐(𝛼i +Φi)

0

…

−𝑙i��i��i 𝑐(𝛽i) 𝑠 (𝛼i +Φi) − 𝑙i��i2 𝑠(𝛽i) 𝑐 (𝛼i +Φi)

+𝑙i��i��i c(𝛽i) 𝑐(𝛼i +Φi) − 𝑙i��i2 𝑠(𝛽i) 𝑐s(𝛼i +Φi)]

(4-82)

where

xH

yH

zH

xP

zP

xLi+1

zLi+1

γHhi+1

hi

pi+1

pi

βi+1

αi+1

Hhi+1

Ppi+1

Hpi+1

ηi/i+1

γP

Figure 4-6: Local actuator reference frame and auxiliary variables [15]

xy

z

rigid bar

p1

p2

p2

p1

u2

u1

p1 cos(u1) = p2 cos(u2)

Figure 4-7: Kinematic constraint of rigid body motion [15]


��i =

��i|L,x

��i|L,x2 +��i|L,y

2 ��i|L,y −��i|L,y

��i|L,x2 +��i|L,y

2 (4-83)

��i =

−1

√1−��i|L,z2

��i|L,z

(4-84)

With that, the platform translational acceleration ��H P|H can be expressed in terms of the auxiliary

variables by computing the time derivative of Eq. (4-76):

��H P|H = 1

3∑ ��H i|H𝑖 (4-85)

where again three platform junction points need to be chosen, which form a triangle centered in 𝐜P.

Choosing 𝑖 = 2,4,6 here, ��H P|H is expressed in terms of the six auxiliary variables ��2, ��4, ��6 and

��2, ��4, ��6. Note that 𝛼2, 𝛼4, 𝛼6 and 𝛽2, 𝛽4, 𝛽6 as well as ��2, ��4, ��6 and ��2, ��4, ��6 can be computed according to the previous two chapters.

In the next step, the platform angular acceleration ��H P|H will be expressed in terms of the auxiliary

variables, too. Therefore, the equation for the platform junction point accelerations derived in Chapter 4.2 is reused:

��H i|H = ��H P|H + ��H P 𝐩P i|P

(4-86)

= ��H P|H + ([ ��

HP|H]x

+ ([ 𝛚H P|H]x)2

) 𝐑H P 𝐩P i|P

= ��H P|H + ([ ��

HP|H]x

+ ([ 𝛚H P|H]x)2

) 𝐩P i|H

Again, Eq. (4-86) can be rearranged in matrix form for 𝑖 = 2,4,6:

𝐂 − ([ 𝛚H P|H]x

)2

𝐏 = [ ��H P|H]x

𝐏 (4-87)

where 𝐂 = [ ��H 2|H ��H 4|H ��6|H] − [ ��HP|H ��H P|H ��H P|H] and 𝐏 = [ 𝐩P 2|H 𝐩P 4|H 𝐩P 6|H].

By inserting Eq. (4-82) and (4-85) as well as the now known angular velocity into this equation, the platform angular acceleration can be expressed in terms of the auxiliary variables, too. As before, all nine scalar equations in Eq. (4-87) need to be implemented as a linear equation system of rank three:

��H P|H = 𝐀(acc)−1 𝐛(acc) (4-88)

Similar to before, six additional constraint equations are required. The first three scalar equations can be found from the time derivative of Eq. (4-80):

( ��H i|H − ��H j|H) ∘ ( 𝐩H i|H − 𝐩P j|H) = −( ��H i|H − ��P j|H)2

(4-89)

with 𝑖 = 2,4,6, 𝑗 = 2,4,6, 𝑖 ≠ 𝑗.

The remaining three constraint equations are determined by first expressing the accelerations of the

other three platform junction points, i.e. ��H i|H for 𝑖 = 1,3,5, in spherical coordinates and then

computing the radial component along the leg axis as derived in Appendix A.2:


��H i|R = 𝑙i − 𝑙i��i2 − 𝑙i��i

2 𝑠(𝛽i) (4-90)

Equating Eq. (4-90) with the projection of the platform junction point acceleration onto the leg axis gives the other three scalar constraint equations:

𝑙i − 𝑙i��i2 − 𝑙i��i

2 𝑠(𝛽i) = ��H i|H ∘ ��i|L (4-91)

where ��H i|H for 𝑖 = 1,3,5 can be expressed in terms of the six auxiliary variables ��2, ��4, ��6 and

��2, ��4, ��6 by inserting Eq. (4-85) and (4-88) into (4-86).

4.4 Actuator Simplified Substitute Model

Figure 4-8 illustrates a linear actuator according to the current design concept. A stepper motor drives a spindle via a reduction gear. The spindle rotation is then converted into a translation of the piston by a roller nut. Similar actuators have been flown on different missions and thus are well known and a validated model is available for them in the Airbus mechanism department. This model is very detailed and includes various effects such as noise due to friction of the gear box and the spindle/nut assembly, hysteresis due to mechanical tolerances and a time delay due to motor inductivity, elasticity and other effects. However, some of these effects are out of the bandwidth of the pointing control system and thus are not relevant for the pointing control simulation. The detailed actuator model that is available needs to be run at very high sampling rates (10kHz) and can thus not be included in the pointing control simulations running at much lower sampling rates (usually around 100Hz). Thus, a simplified substitute model has been derived in close cooperation with the Airbus mechanism department that includes the step quantization of the motor, a non-oscillating PT2 element that represents the main effect of the motor dynamics, i.e. inductivity, as well as a linearized representation of the hysteresis curve. Figure 4-9 shows a block diagram of the simplified model.

roller nutspindle

linear guidancestepper motor

gear box

elastic coupling

piston

Figure 4-8: Hexapod linear actuator design concept

k k

quantization

TF

hysteresis non-oscillating PT2 gear ratio pitch per step

step count

actuator length

Figure 4-9: Linear actuator simplified substitute model

Figure 4-10 shows the hysteresis curve of the detailed model in comparison to the linearized representation of it. Figure 4-11 shows the step response of the detailed model in comparison to the simplified model only including a simple, non-oscillating PT2 element. Note that the step response of the simplified model does not depend on the external load and the model is thus only valid if the actuator is well dimensioned for its application and it is not operated close to the boundaries of its operational envelope.


(a) (b)

Figure 4-10: Actuator hysteresis curve of (a) detailed model and (b) simplified substitute model

Figure 4-11: Actuator step response of detailed model and simplified substitute model

55

5 Spacecraft Attitude Dynamics with

Hexapod in the Loop Spacecraft Attitude Dynamics with Hexapod in the Loop

5.1 Overview

The MAM represents a significant part of the total SC mass. Thus, with the MAM moving relative to the SC, the system cannot be considered as one rigid body anymore. In general, the system now has twelve degrees of freedom – three rotational plus three translational DoF of the whole system in inertial space and three rotational plus three translational DoF of the MAM relative to the SC. Such a system can be represented by two rigid bodies in free space, which are connected for example by a set of linear actuators in series with spring-damper systems as illustrated in Figure 5-1 (a). Given the forces applied by the linear motors, the equations of motion of the combined system could be formulated in twelve DoF. However, to meet the high pointing performance requirements, the hexapod mechanism needs to be design stiff enough such that any flexible modes between the two bodies can be neglected for pointing performance analysis. Thus, for simplicity the hexapod actuators are assumed to be infinitely stiff, leading to a system of two rigid bodies in free space with relative prescribed motion as illustrated in Figure 5-1 (b). The term prescribed motion is introduced in [28] in the context of controlled joint variables in multibody systems. As described there, if a joint with one degree of freedom is controlled such that its motion can be considered as prescribed, the degree of freedom of the entire system is reduced by one. Thus, with the six hexapod actuator legs being controlled, the new degree of freedom of the entire system is six.

(a) xI

yI

zI

c1

c2

(b) xI

yI

zI

c1

c2

Figure 5-1: System of two rigid bodies connected by linear actuator legs (a) with and (b) without flexible modes

Hereafter, the equations of motion are first derived for a general two-body system with prescribed relative motion in free space in Chapter 5.2. In Chapter 5.3 the results are then formulated for the problem at hand, i.e. the spacecraft attitude dynamics with prescribed hexapod motion. Finally, the

56 5. Spacecraft Attitude Dynamics with Hexapod in the Loop

inertia variations of the combined system due to the prescribed motion of the hexapod and the disturbance torque as seen by the SC attitude controller are analyzed and the virtual forces in the actuator legs are computed.

5.2 Two-Body System with Prescribed Relative Motion in Free Space

In the following derivations, the formalism and nomenclature introduced in Chapter 2.5 for the Newton-Euler formulation of multi-body dynamics is applied. Again, though not explicitly indicated by subscript |I for better readability, all quantities in this chapter are expressed in inertial reference frame if not indicated differently. Consider the simple two body system described in the previous chapter and replace the six linear 1-DoF actuators by one fictitious 6-DoF actuator as Figure 5-2 to simplify the following derivations. Assume that the two bodies are connected by a massless rod and let 𝐫c2,1 = 𝟎,

i.e. the connection point is in 𝐜2. With both bodies of the system being free ends, there are no reaction forces of the environment, such that 𝐅0,1 = 𝟎, 𝐅2,3 = 𝟎 as well as 𝐌0,1 = 𝟎, 𝐌2,3 = 𝟎.

The equations of motion for body 1 are then defined as follows:

��1 = 𝐅1 − 𝐅1,2 (5-1)

��1 = 𝐌1 −𝐌1,2 − 𝐫c1,2 × 𝐅1,2 (5-2)

Remembering that 𝐫c2,1 = 𝟎, the equations of motion for body 2 are:

��2 = 𝐅2 + 𝐅1,2 (5-3)

��2 = 𝐌2 +𝐌1,2 − 𝐫c2,1 × 𝐅1,2

(5-4) = 𝐌2 +𝐌1,2

If the relative motion between bodies 1 and 2 is prescribed, e.g. by six linear 1-DoF actuators as described previously or by one fictitious 6-DoF actuator, the total system has six degrees of freedom and can be fully described by position and orientation of any of the two rigid bodies in the inertial reference frame. Hereafter, the equations of motion are expressed in terms of position and orientation of body 1. Therefore Eq. (5-3) and (5-4) are rearranged as follows in order to eliminate the coupling forces in Eq. (5-1) and (5-2).

−𝐅1,2 = 𝐅2 − ��2 (5-5)

−𝐌1,2 = 𝐌2 − ��2 (5-6)

In preparation for the upcoming application, it is further assumed, that the external force 𝐅2 and

torque 𝐌2 on body 2 are zero and thus 𝐅1,2 = ��2 and 𝐌1,2 = ��2 are inserted into Eq. (5-1) and (5-2).

The equations of motion of the two-body system are thus:

��1 = 𝐅1 − ��2 (5-7)

��1 = 𝐌1 − ��2 − 𝐫c1,2 × ��2 (5-8)

As mentioned in the beginning of this chapter, all quantities are expressed in inertial reference frame {I} and thus linear momentum and angular momentum are defined as follows:

𝐐i = 𝐐i|I = 𝓘i(tra) 𝐯I i|I (5-9)

5. Spacecraft Attitude Dynamics with Hexapod in the Loop 57

𝐇i = 𝐇i|I = 𝓘i(rot)|I 𝛚I

i|I (5-10)

where 𝐯I i|I and 𝛚I i|I are the translational and angular velocity of body 𝑖 w.r.t. and expressed in {I}-

frame, 𝓘i(tra) is the translational inertia matrix and 𝓘i(rot)|I is the rotational inertia matrix of the body

expressed in {I}-frame.

rc ,21rc ,21

xI

yI

zI

c1

c2

Figure 5-2: System of two rigid bodies in free space

With that and under the following previously introduced assumptions:

1. Massless infinitely stiff 6-DoF actuator with infinite actuation capacity connecting the two bodies

2. Prescribed relative motion of the two bodies

3. No external forces and torques on body 2

the equations of motion of the two-body system can be expressed as follows:

𝓘1(tra) ��I 1|I = 𝐅1|I − 𝐅2(kin)|I (5-11)

𝓘1(rot)|I ��I1|I = 𝐌1|I −𝐌2(kin)|I − 𝐫c1,2|I × 𝐅2(kin)|I − ��1(rot)|I 𝛚

I1|I (5-12)

Or expressed in the body frame {B1} of body 1:

𝓘1(tra) ��I 1|B1 = 𝐅1|B1 − 𝐑B1I𝐅2(kin)|I (5-13)

𝓘1(rot)|B1 ��I 1|B1 = 𝐌1|B1 − 𝐑B1I𝐌2(kin)|I − 𝐑B1

I(𝐫c1,2|I × 𝐅2(kin)|I) −

[ 𝛚I 1|B1]x𝓘1(rot)|B1 𝛚I 1|B1 (5-14)

with the kinetic force 𝐅2(kin)|I and torque 𝐌2(kin)|I of body 2 being functions of its motion in inertial

space, which is given by the motion of body 1 relative to {I}-frame as well as the prescribed relative motion between the two bodies.

𝐅2(kin)|I = ��2|I = 𝓘2(tra) ��I 2|I (5-15)

𝐌2(kin)|I = ��2|I = 𝓘2(rot)|I ��I

2|I + ��2(rot)|I 𝛚I

2|I (5-16)

where the linear and angular velocities of body 2 in inertial space are defined by the prescribed relative motion between body 2 and body 1 and the inertial motion of body 1:

𝐯I 2|Ix = 𝐯1 2|I + 𝐑I B1[ 𝛚I1|B1]x

𝐫c1,c2|B1 + 𝐯I 1|I (5-17)

𝛚I 2|I = 𝐑I B2 𝛚1 2|B2 + 𝐑I B1 𝛚I 1|B1 (5-18)


5.3 Spacecraft Attitude Dynamics with Prescribed Hexapod Motion

Assuming mass-less actuator legs and prescribed motion of the hexapod platform, i.e. perfect guidance tracking, the combined system of the spacecraft and the hexapod can be reduced to the simple two-body system discussed in the previous chapter, where body 1 is the spacecraft (focal plane module, service module and telescope structure) and body 2 is the hexapod platform as illustrated in Figure 5-3.

xI

yB

zI xB

zB

xH

yH

zH

xP zP

HcP

BcP

extensible leg actuator

hexapod base

hexapod platform

combined system center of mass

yI

IcB

SC center of mass without platform

platform center of mass

Figure 5-3: Two-body system formed by spacecraft and hexapod platform

The system thus has six degrees of freedom, i.e. position and orientation of the combined system in space. To express the equations of motion of this system as given in Eq. (5-13) and (5-14), the kinematic force and torque of the hexapod platform due to its inertial motion need to be derived.

The linear momentum of the moving platform due to the velocity 𝐯P|II relative to and expressed in

inertial {I}-frame is given by the following expression:

𝐐P|I = 𝓘P(tra) 𝐯P|II (5-19)

where 𝐯P|II = ��P|I

I and 𝓘P(tra) is the translational inertia matrix of the platform. Under the

assumption, that the geometric center point of the platform is also its center of mass, the translational inertia matrix simplifies as given hereafter. If this is not the case, great care must be taken that the inertia matrix is always expressed in the correct reference frame. As this only complicates the following derivations in writing, the simplified case is assumed hereafter:

𝓘P(tra) = 𝕀3x3𝑚P

with 𝑚P being the platform mass. The position of the platform center point 𝐜P|II relative to and

expressed in {I}-frame can be computed as follows

𝐜P|II = 𝐑I B( 𝐜B H|B + 𝐑B H 𝐜P|H

H ) + 𝐜B|II (5-20)

where 𝐜B H|B is the time-constant position of the hexapod base relative to and expressed in SC body

frame {B}, 𝐑B H is the time-constant rotation matrix from {H}- to {B}-frame, 𝐜P|HH is the time-varying


position of the platform center relative to and expressed in {H}-frame, 𝐜B|II is the time-varying

position of the SC center of mass relative to and expressed in {I}-frame and 𝐑I B is the time-varying rotation matrix of the SC body frame {B} in the inertial frame. Figure 5-3 further illustrates these relations. The velocity and acceleration of the platform derived in inertial frame are then given as follows:

��P|II = 𝐑I B 𝐑B H ��P|H

H + 𝐑I B[ 𝛚I

B|B]x𝐜P|BB + ��B|I

I (5-21)

��P|II = 𝐑I B (([ 𝛚

IB|B]x

2+ [ ��I B|B]x

) 𝐜P|BB + 2[ 𝛚I B|B]x

𝐑B H ��P|HH +

𝐑B H ��P|HH ) + ��B|I

I (5-22)

In Eq. (5-21) the first summand is the translation of the platform due to the active movement relative to the base and expressed in {I}-frame here. This movement is controlled by the six linear actuators of the hexapod and is thus called ‘active’. The other two summands are the translation of the platform due to the passive movement of the platform in inertial space. This movement is caused by the translation of the SC in inertial space plus the rotation of the SC around its center of mass via the lever

arm 𝐜P|BB as illustrated in Figure 5-3 and is thus called ‘passive’.

Inserting Eq. (5-21) into (5-19) gives the linear momentum of the platform:

𝐐P|I = 𝓘P(tra) 𝐑I B 𝐑B H ��P|HH + 𝓘P(tra) ( 𝐑I B[ 𝛚

IB|B]x

𝐜P|BB + ��B|I

I ) (5-23)

where the left-hand summand is the linear momentum due to active movements of the platform as derived in [18], converted to inertial frame, and the right-handed summand is the additional linear momentum of the platform due to passive movements caused by SC rotation and translation.

Computing the time derivative of Eq. (5-23), where ��P(tra) = 𝟎, gives the kinetic force vector that acts

on the platform due to translational motion:

𝐅P(kin)|I = ��P|I = 𝓘P(tra) ��P|II

(5-24)

= [𝓘P(tra) 𝐑I B[ 𝛚I

B|B]x2𝐜P|BB ] + [𝓘P(tra) 𝐑I B2[ 𝛚

IB|B]x

𝐑B H ��P|HH ] +

[𝓘P(tra) 𝐑I B ([ ��I

B|B]x𝐜P|BB + 𝐑B H ��P|H

H ) + 𝓘P(tra) ��B|II ]

where the first summand is the centripetal force, the second the Coriolis force and the third the Euler and inertial force due to the linear acceleration of the platform.

The angular momentum of the platform about its center of mass is given as follows:

𝐇P|I = 𝓘P(rot)|I 𝛚I

P|I

(5-25) = 𝐑I P𝓘P(rot)|P 𝛚I P|P

The {P}-frame is fixed relative to the platform and its axes coincide with the principal directions of the platform’s inertia. Thus, the rotational inertia matrix of the platform is time-constant and diagonal, if expressed in {P}-frame:

𝓘P(rot)|P =

[

ℐPxx 0 00 ℐPyy 0

0 0 ℐPzz

]

(5-26)


The inertia matrix can be transformed to {I}-frame as derived in [9]:

𝓘P(rot)|I = 𝐑I P𝓘P(rot)|P 𝐑P I (5-27)

The angular velocity 𝛚I P|P of the platform relative to the {I}-frame in Eq. (5-25) can be determined as

follows:

𝑑

𝑑𝑡𝐑P I = 𝑑

𝑑𝑡( 𝐑P H 𝐑H B 𝐑B I)

(5-28) [ 𝛚I P|P]x

T𝐑P I

= [ 𝛚H P|P]xT𝐑P I + 𝐑P B[ 𝛚

IB|B]x

T𝐑B I

Right-multiplying both sides of Eq. (5-28) with 𝐑I P and using [… ]xT = −[… ]x leads to:

−[ 𝛚I P|P]x = −[ 𝛚H P|P]x

− 𝐑P B[ 𝛚I

B|B]x𝐑B P (5-29)

As derived in [9] it is [ 𝐑2 1a1]x = 𝐑2 1[a1]x 𝐑1 2. Applying this relation to Eq. (5-29) leads to:

[ 𝛚I P|P]x = [ 𝛚H P|P]x

+ [ 𝐑P B 𝛚I B|B]x (5-30)

With all terms in Eq. (5-30) being skew-symmetric matrices, comparing the matrix elements of the left and the right side of the equation finally gives:

𝛚I P|P = 𝛚H P|P + 𝐑P B 𝛚I B|B (5-31)

with

𝐑P B = 𝐑P H 𝐑H B

and

𝛚H P|P = 𝐑P H 𝛚H P|H = 𝐑P H𝐉A��P(rot)|E

The time derivative of Eq. (5-31), which will be required in the next steps, is given by:

��I P|P = ��H P|P + ��P B 𝛚I B|B + 𝐑P B ��I B|B (5-32)

with

��H P|P = ��P H𝐉A��P(rot)|E + 𝐑P H��A��P(rot)|E + 𝐑P H𝐉A��P(rot)|E

and

��P B = ��P H 𝐑H B

Inserting Eq. (5-31) into Eq. (5-25) gives the angular momentum in inertial frame:

𝐇P|I = 𝐑I P𝓘P(rot)|P 𝛚I P|P

(5-33) = 𝐑I P𝓘P(rot)|P 𝛚H P|P + 𝐑I P𝓘P(rot)|P 𝐑P B 𝛚I B|B


Note that the left-hand term in Eq. (5-33) is the angular momentum due to the active movement of the platform relative to the {H}-frame caused by controlled actuator movements as derived in [18]. The right-hand term is the angular momentum of the platform caused by the rotation of the SC. If expressed in an inertial frame, the time derivative of the angular momentum equals the torque. Thus,

the kinetic torque around 𝐜P applied to the platform can be found by computing ��P|I, where

��P(rot)|P = 𝟎:

𝐌P(kin)|I = ��P|I

(5-34)

= 𝐑I P[ 𝛚H

P|P + 𝐑P B 𝛚I B|B]x𝓘P(rot)|P( 𝛚H P|P + 𝐑P B 𝛚I B|B) +

𝐑I P𝓘P(rot)|P( ��H P|P + ��P B 𝛚I B|B + 𝐑P B ��I B|B)

Inserting Eq. (5-24) into (5-13) finally leads to the translational equation of motion of the two-body system:

𝓘B(tra) ��I B|B = 𝐅B|B − 𝐑B I𝐅P(kin)|I

(5-35)

= 𝐅B|B − [𝓘P(tra)[ 𝛚I

B|B]x2𝐜P|BB ] − [𝓘P(tra)2[ 𝛚

IB|B]x

𝐑B H ��P|HH ] −

[𝓘P(tra) (([ ��I

B|B]x𝐜P|BB + 𝐑B H ��P|H

H ) + ��B|BI )]

Solving Eq. (5-35) for ��I B|B provides the translational acceleration of the SC, which can then be

integrated twice, giving the translational velocity and position of the SC:

��I B|B = (𝓘(tra))−1(𝐅B|B − 𝐅P(kin)(cp)|B − 𝐅P(kin)(cor)|B − 𝐅P(kin)(inrt)|B) (5-36)

where 𝓘(tra) is the translational inertia matrix of the complete system, 𝐅B|B is the sum of all external

forces, 𝐅P(kin)(cp)|I is the centripetal force of the hexapod platform mass due to SC rotation,

𝐅P(kin)(cor)|B is the Coriolis force and 𝐅P(kin)(inrt)|B is the inertial force due to the linear acceleration

of the platform mass in inertial space:

𝓘(tra) = 𝓘B(tra) + 𝓘P(tra)

𝐅P(kin)(cp)|B = [𝓘P(tra)[ 𝛚I

B|B]x2𝐜P|BB ]

𝐅P(kin)(cor)|B = 2 [𝓘P(tra)[ 𝛚I

B|B]x𝐑B H ��P|H

H ]

𝐅P(kin)(inrt)|B = 𝐅P(kin)(tra)|B + 𝐅P(kin)(rot)|B = 𝓘P(tra) [ 𝐑B

H ��P|HH + [ ��I B|B]x

𝐜P|BB ]

Applying 𝐑(𝐚 × 𝐛) = 𝐑𝐚 × 𝐑𝐛 for 𝐑 being a rotation matrix to Eq. (5-14) and inserting Eq. (5-34) finally leads to the rotational equation of motion of the two-body system:

𝓘B(rot)|B ��I |B = 𝐌B|B − 𝐑B I𝐌P(kin)|I − 𝐜B P|B × 𝐑B I𝐅P(kin)|I −

[ 𝛚I B|B]x𝓘B(rot)|B 𝛚I B|B (5-37)

As can be seen above, Eq. (5-37) includes the translational acceleration ��B|BI . Thus, translation and

rotation are coupled. Therefore, inserting Eq. (5-36) into Eq. (5-37) and solving for ��I |B provides the

angular acceleration of the SC, which can then be integrated twice, giving the angular velocity and orientation of the SC:


��I |B = (𝓘(rot)|B)−1(𝐌B|B +𝐌D|B − [ 𝛚

IB|B]x

𝓘(rot)|B 𝛚I B|B) (5-38)

where 𝓘(rot)|B is the rotational inertia matrix of the complete system, which is further discussed in the

next chapter, and 𝐌D|B is the disturbance torque caused by the prescribed motion of the hexapod

platform.

𝓘(rot)|B = 𝓘B(rot)|B + 𝓘P(rot)|B − 𝓘P(tra)𝓘B(tra)𝓘(tra)−1 [ 𝐜B P|B]x

2

𝐌D|B = − 𝐑B P [[ 𝛚H

P|P]x𝓘P(rot)|P( 𝛚H P|P + 𝐑P B 𝛚I B|B) +

[ 𝐑P B 𝛚I B|B]x𝓘P(rot)|P 𝛚H P|P] − 𝐑B P𝓘P(rot)|P( ��H P|P +

��P B 𝛚I B|B) − [ 𝐜BP|B]x

[(𝓘(tra))−1𝐅B|B + (𝕀3x3 −

𝓘(tra)−1 )(𝐅P(kin)(cor)|B + 𝐅P(kin)(tra)|B)]

5.3.1 Hexapod Induced Inertia Variation

The inertia matrix of the combined system, derived in the last chapter can be rewritten as follows for better interpretation of the terms:

𝓘(rot)|B = 𝓘B(rot)|B − 𝓘B(tra) [−

𝑚P

𝑚P+𝑚B𝐜B P|B]

x

2+ 𝓘P(rot)|B −

𝓘P(tra) [𝑚𝑆


x

2

(5-39)

where the second term is the Steiner part of the spacecraft with respect to the center of mass of the

combined system with −𝑚P

𝑚P+𝑚B𝐜B P|B being the vector from the center of mass of the combined

system to the center of mass of the spacecraft without the hexapod platform, i.e. the origin of the {B}-frame. Correspondingly, the last term is the Steiner part of the hexapod platform with respect to the center of mass of the combined system. Hereafter it is shown that these two Steiner parts are equivalent to the last term in the formulation for 𝓘(rot)|B derived in the previous chapter:

𝓘B(tra) [−

𝑚P


x

2+ 𝓘P(tra) [

𝑚𝑆


x

2

= 𝕀3x3𝑚𝑆 (𝑚P

𝑚P+𝑚B)2[ 𝐜B P|B]x

2+ 𝕀3x3𝑚P (

𝑚B

𝑚P+𝑚B)2[ 𝐜B P|B]x

2

= 𝕀3x3𝑚P2𝑚B+𝑚P𝑚B

2

(𝑚P+𝑚B)2 [ 𝐜B P|B]x

2= 𝕀3x3

𝑚P𝑚B

𝑚P+𝑚B[ 𝐜B P|B]x

2= 𝓘P(tra)𝓘B(tra)𝓘(tra)

−1 [ 𝐜B P|B]x2

5.3.2 Leg Actuator Forces

Note that the coupling force and torque between the spacecraft and hexapod platform, derived in the previous chapter, is applied to the platform by six forces acting at the platform junction points in the direction of the actuator axes. These coupling force and torque can be assembled in a generalized force vector expressed in {H}-frame as follows:

𝐟P(kin)|H = [𝐅P(kin)|H𝐌P(kin)|H

] (5-40)


This generalized force is applied to the platform by the six actuators and is thus the vector sum of the six forces along the actuator axes and the vector sum of the corresponding moments around the platform center of mass:

𝐟P(kin)|H = [

��1|H … ��6|H

[ 𝐑H P 𝐩P 1|P]x��1|H … [ 𝐑H P 𝐩P 6|P]x

��6|H] 𝛕P

(5-41) = 𝐉CT𝛕P

where 𝛕P = [𝛕𝟏 … 𝛕𝟔]𝐓 is the actuator forces vector with τi being the force applied to the platform

by actuator 𝑖 along its axial direction ��i|H. 𝐉C is the Jacobian matrix already known from Chapter 4.1.

Thus, the actuator forces can be computes from the generalized force vector using the inverse Jacobian as follows:

𝛕P = 𝐉C−T𝐟P(kin)|H (5-42)

65

6 Pointing System State Determination with

Hexapod in the Loop Pointing System State Determination with Hexapod in the Loop

6.1 Overview

Three quantities are required for accurate pointing control in the Athena mission. First, the LoS orientation in inertial reference frame needs to be known with high accuracy for the reconstruction of the point spread function as described in Chapter 1.2.1. Therefore, the STR is directly mounted on the MAM and measures the orientation of the MAM in inertial space. Second, the SC attitude in inertial reference frame needs to be known, which is the feedback in the closed-loop attitude control. As it is not directly measured, it needs to be reconstructed from the STR measurement and the best knowledge of the MAM orientation relative to the SC. Third, the MOA to LoS misalignment, i.e. the angle between the LoS and the mirror optical axis, needs to be known in order to compensate it with small hexapod maneuvers. To prevent vignetting, the LoS and MOA are required to be aligned during the observation. The MOA to LoS misalignment will be denoted simply as MOA misalignment hereafter. As mentioned before, the LoS is defined by two points, the MAM node and the center point of the detector. It thus depends on the detector selection and the position of the MAM relative to the SC. The detector selection is a known parameter. The LoS orientation relative to the SC can thus be reconstructed from a known position of the MAM as derived in Appendix A.3. The mirror optical axis is the perpendicular of the MAM through its node and depends on the orientation of the MAM relative to the SC. Both, LoS and MOA orientation relative to the SC are thus functions of the MAM position and orientation respectively. The MAM position and orientation relative to the SC is defined by its position and orientation relative to the hexapod base plus the position and orientation of the hexapod base relative to the SC. The lather is influenced by thermal distortions of the telescope structure, leading to small position and orientation variations of the hexapod base relative to the FPM. The ESA design concept anticipates some kind of on-board metrology that can be used to measure and thereafter compensate the above mentioned thermal distortions and the resulting MOA misalignment. Two different approaches are discussed for the OBM hereafter: The ‘MAM Absolute Pose Measurement Based Approach’ and the ‘MOA Misalignment Measurement Based Approach’.

6.2 MOA Misalignment Measurement Based Approach

This approach is based on an OBM technology that directly measures the error angle between the LoS and the MOA, i.e. the MOA misalignment.With the OBM measuring only the MOA misalignment, no information about the orientation of the MAM relative to the SC is available. However, this information is required to reconstruct the SC attitude from STR measurements. Therefore, an additional Hexapod State Metrology (HSM) is required that provides information about the hexapod platform orientation relative to the hexapod base. Neglecting the thermal distortions between the hexapod base and the detectors on the FPM, this knowledge can be used to reconstruct the SC attitude. Two different approaches for the HSM are discussed in Chapter 6.2.4, one of which is purely based on software algorithms and requires no additional hardware. The complete state determination system with this OBM approach is described in Chapter 6.2.1 hereafter. The corresponding knowledge errors are analyzed in Chapter 6.2.2.

66 6. Pointing System State Determination with Hexapod in the Loop

6.2.1 State Determination System Overview

Figure 6-1 shows a simplified block diagram of the state determination system with an OBM that directly measures the MOA misalignment. Starting at the top left of the diagram, the MAM platform

position 𝐜PH and orientation 𝐑H

P relative to the hexapod base frame are available as outputs of the forward kinematic block. These quantities are measured by the HSM. Adding the nominal position

𝐜HB0 and orientation 𝐑B0

H of the hexapod base frame to the HSM measurements provides the best

knowledge of the MAM platform position ��PB and orientation ��B

P in SC body frame. Adding the nominal position and orientation of the hexapod base frame relative to SC body frame plus the small

position and orientation changes due to thermal distortions, 𝛿𝐜B0B and 𝛿𝐑B

B0 , to the outputs of the forward kinematic leads to the true position and orientation of the MAM platform relative to the SC

body frame. The LoS orientation 𝐑BLoS is reconstructed from the true position of the MAM platform

relative to SC body. The MOA misalignment 𝐑PLoS is computed as the difference between the LoS

orientation and the platform orientation relative to the SC body frame. This quantity is measured by

the OBM. The MAM platform orientation in inertial space, 𝐑IP , is computed from the output of the

dynamics block, i.e. the SC attitude in inertial space 𝐑IB , and the MAM platform orientation relative

to the SC body frame 𝐑BP . This quantity is measured by the STR. Note that dotted line to the right of

the OBM and STR blocks illustrates that the OBM measurement could be used to compensate the error due to thermal distortions in the LoS orientation in inertial space (cf. knowledge errors with and without taking the OBM measurements into account at this point as discussed in Chapter 6.2.2).

hex. fwd. kinematic

S/C dynamic with prescr. Hex motion

δ RB δcB0

RH

l

cP

τ

H

P

A*B

RB0

H

+

cH

B0

A*B +

LoS reconstr.

BB0

RB

P cP

B

A*BT

RB

LoS

OBM

A*BR I

B

RP

LoS

RP

LoS~

STRR I

P

AT*B

A*B

R I

P~

RI

LoS~

R I

B

HSMA*B

+cH

B0

RB0

H

RH

P~

cP

H~

RB

P~

cP

B~

~

Figure 6-1: MOA misalignment measurement based state determination approach

The measured quantities mentioned above are determined by STR, OBM and HSM as follows:

Determination of MOA misalignment

The MOA misalignment is directly measured by the OBM.

Determination of SC attitude in inertial space

The STR measures the orientation of the MAM platform in inertial space, i.e. ��IP . The orientation of

the MAM platform relative to the SC is not measured by the OBM in this approach. Instead, the orientation of the MAM platform relative to the hexapod base is determined by the HSM. With no

6. Pointing System State Determination with Hexapod in the Loop 67

dedicated knowledge about the orientation change between hexapod base and SC body due to thermal distortions of the telescope structure, subtracting the nominal orientation of the hexapod base relative to the SC provides the best estimate of the MAM orientation relative to the SC:

��BP = ��H

P 𝐑B0H (6-1)

The SC attitude in inertial frame is then given by subtracting Eq. (6-1) from the STR measurement:

��IB = ��B

TP ��IP (6-2)

Note that this approach does not account for the thermal distortions of the telescope, which are therefore included as measurement error in the SC attitude measurement, as will be shown in the next chapter.

Determination of LoS orientation in inertial space

If the LoS and MOA are perfectly aligned after the corrective maneuvers with the hexapod, the STR directly measures the orientation of the LoS in inertial space:

��ILoS = ��I

P (6-3)

If this assumption is not completely true, the remaining MOA misalignment directly contributes to the LoS knowledge error. This contribution to the LoS knowledge error could be decreased by frequently performing OBM measurements during the observation and thus decreasing the impact of the thermal distortions on the LoS knowledge:

��ILoS = 𝛿��P

LoS ��IP (6-4)

6.2.2 Knowledge Errors

The following measurements are available as described above with the corresponding measurements errors. First, the STR measurement of the MAM orientation in inertial space:

��IP = 𝛿𝐑STR 𝐑I

P ≈ (𝕀3x3 − [δ𝛈STR]x) 𝐑IP (6-5)

where 𝛿𝐑STR is a rotation matrix representing the estimation errors around the {P}-frame axes. Its linear approximation for small angles is given by:

𝛿𝐑STR ≈ (𝕀3x3 − [δ𝛈STR]x) (6-6)

with δ𝛈STR = [𝛿𝜂STR,x 𝛿𝜂STR,y 𝛿𝜂STR,z]T being a vector of small measurement errors around the {P}-frame axes.

Second, the OBM measurement of the MOA to LoS misalignment:

��PLoS = 𝐑P

LoS δ𝐑OBM ≈ 𝐑PLoS (𝕀3x3 − [δ𝛈OBM]x) (6-7)

with δ𝛈OBM = [𝛿𝜂OBM,x 𝛿𝜂OBM,y 𝛿𝜂OBM,z]T being a vector of small measurement errors around the {P}-frame axes.

Third, the HSM measurement of the hexapod platform pose relative to the hexapod base:

��HP = 𝐑H

P δ𝐑HSM ≈ 𝐑HP (𝕀3x3 − [δ𝛈HSM]x) (6-8)


with δ𝛈HSM = [𝛿𝜂HSM,x 𝛿𝜂HSM,y 𝛿𝜂HSM,z]T being a vector of small measurement errors around the {H}-frame axes.

With these measurements, the knowledge errors on the previously defined quantities of interest are determined as follows.

MOA to LoS misalignment

The MOA misalignment is directly measured by the OBM and thus the knowledge error is equal to the OBM measurement error:

��PLoS 𝐑P

TLoS = 𝐑PLoS (𝕀3x3 − [δ𝛈OBM]x) 𝐑P

TLoS

(6-9)

= 𝕀3x3 − 𝐑PLoS [δ𝛈OBM]x 𝐑LoS

P⏟

OBM noise added in {P}−frame

SC attitude in inertial space

The SC attitude is reconstructed from the STR measurement and the best estimate of the MAM orientation relative to the SC. The knowledge error thus includes the STR measurement error, HSM measurement error and the unknown rotation of the hexapod base frame relative to its nominal orientation due to thermal distortions:

��IB 𝐑I

TB = 𝐑HB0 ��H

TP⏟HSM

��IP⏟STR

𝐑ITB

(6-10)

= 𝐑H

B0 ( 𝐑HP (𝕀3x3 − [δ𝛈HSM]x))

T((𝕀3x3 − [δ𝛈STR]x) 𝐑I

P ) 𝐑ITB

= 𝛿𝐑BB0⏟ thermal

distortions(

𝕀3x3 − 𝐑PB [δ𝛈STR]x 𝐑B

P⏟ STR noise added in

{P}−frame

− 𝐑HB [δ𝛈HSM]x

T 𝐑BH

⏟ HSM noise added in

{H}−frame )

LoS orientation in inertial space

Without the OBM measurement taken into account, the STR measurement is directly used as measurement of the LoS orientation in inertial space. The knowledge error then includes the STR measurement error as well as the MOA misalignment:

��ILoS 𝐑I

TLoS = ��IP⏟STR

𝐑ITLoS

(6-11)

= (𝕀3x3 − [δ𝛈STR]x) 𝐑IP 𝐑LoS

I

=

(

𝕀3x3 − [δ𝛈STR]x⏟ STR noise added in

{P}−frame )

𝐑LoSP⏟ MOA

misalign.

With the OBM measurement of the MOA misalignment taken into account, the knowledge error of the LoS orientation in inertial space does include the STR measurement error as well as the OBM measurement error but not the MOA misalignment anymore:


��ILoS 𝐑I

TLoS = ( ��PLoS⏟ OBM

��IP⏟STR

) 𝐑ITLoS

(6-12)

= ( 𝐑PLoS (𝕀3x3 − [δ𝛈OBM]x)(𝕀3x3 − [δ𝛈STR]x) 𝐑I

P ) 𝐑LoSI

=

(

𝕀3x3 − 𝐑PLoS [δ𝛈STR]x 𝐑LoS

P⏟

STR noise added in{P}−frame

− 𝐑PLoS [δ𝛈OBM]x 𝐑LoS

P⏟

OBM noise added in{P}−frame )

6.2.3 On-Board Metrology

The angle between the MOA and the LoS can for example be measured by the displacement of a laser beam pointing from the MAM towards the FPM. For small angles, this displacement can be used as a measurement of the misalignment angle. To be included in the simulations, a simplified model of this form of OBM represents the measurement errors on the displacement measurement by band-limited white noise with no correlation between the axes.

6.2.4 Hexapod State Metrology

Two different HSM technologies are discussed and compared hereafter. The first approach is a purely software based solution. Under the assumption, that each commanded step is executed properly by the hexapod linear actuators, their lengths can be determined by counting the commanded steps from a known initial state. Through the forward kinematic of the hexapod, the hexapod states can be reconstructed from the then known actuator lengths. The advantages of this approach are that it is a purely software based solution with no additional hardware required, it provides accurate and reliable position knowledge at the end of a maneuver. The disadvantages are the knowledge errors during a maneuver due to discrete step count compared to non-discrete execution of a step by the motors and the dependence of the knowledge accuracy on the accuracy of the motor step execution (mechanical tolerances, etc.).

The second approach anticipates the use of optical encoders that directly measure the actuator lengths. This also provides knowledge of the velocity and acceleration of the actuator during the execution of a step if the sampling frequency of the optical encoders is high enough. Again, the hexapod states can be reconstructed using the forward kinematic of the hexapod. However, to the best knowledge of the author, no previous European mission with linear optical encoders used in similar linear actuators exists. Thus, this approach requires new technology developments in order to adapt space qualified encoders used in other missions for integration into the Athena hexapod actuators. An example of optical encoders used for precision pointing applications in space is given in [29] for the Laser Communication Terminal. The advantages of this approach are that the direct measurement of the actuator length provides a better knowledge of the hexapod states during the maneuver and that not only length, but also velocity and acceleration information is available. The disadvantages are that additional hardware and corresponding development efforts are required.

Based on the fact that for the currently anticipated use of the hexapod accurate and reliable knowledge of the hexapod states is only required at the end of a maneuver, step count can be seen as the preferable solution.

6.3 MAM Absolute Pose Measurement Based Approach

This approach is based on an OBM technology that measures the absolute position and orientation of the MAM platform relative to the SC body frame, including all thermal distortions and other error sources. A design concept for the corresponding technology is briefly discussed Chapter 6.3.3. The


complete state determination system with this OBM approach is described in Chapter 6.3.1 hereafter. The corresponding knowledge errors are analyzed in Chapter 6.3.2

6.3.1 State Determination System Overview

Figure 6-2 shows a simplified block diagram of the state determination system with an OBM that measured the absolute pose of the MAM platform. Note that in this case no Hexapod State Metrology

is required and the position 𝐜PB and orientation 𝐑B

P of the platform in SC body frame are directly measured by the OBM. The rest of the state determination system remains the same as in the previously described approach.

hex. fwd. kinematic

S/C dynamic with prescr. Hex motion

δ RB δcB0

RH

l

cP

τ

H

P

A*B

RB0

H

+

cH

B0

A*B +

LoS reconstr.

BB0

RB

P cP

B

OBM

A*BR I

B

RP

LoS~

GSER I

P

AT*B

A*B

R I

P~

RI

LoS~

R I

B

A*BT

cP

B

RB

P

~

~

RB

LoS~

~

Figure 6-2: Absolute pose measurement based state determination approach

The measured quantities mentioned above are determined by STR, OBM and HSM as follows:

Determination of MOA misalignment

The OBM measures the position and orientation of the MAM relative to the SC. The LoS orientation relative to the SC is reconstructed from the measured MAM position as described in Appendix A.3. The MOA misalignment angle is then computed as follows:

��PLoS = ��B

LoS ��BTP (6-13)

Determination of SC attitude in inertial space

The STR measures the orientation of the MAM platform in inertial space, i.e. ��IP . As before, the SC

attitude is compute by subtracting the best knowledge of the MAM orientation relative to the SC from the STR measurement, only this time it is directly measured by the OBM including thermal distortions of the structure:

��IB = ��B

TP ��IP (6-14)

Determination of LoS orientation in inertial space

The determination of the LoS orientation in inertial space is the same as previously described in Eq. (6-3) and (6-4).


6.3.2 Knowledge Errors

The following measurements are available as described above with the corresponding measurements errors: First, the STR measurement of the MAM orientation is the same as before and defined in Eq. (6-4). Second, the OBM measurement of the MAM platform position:

��P|PB = ��P|P

D − 𝐜B|PD = ( 𝐜P|P

D + δ𝛈OBM,pos) − 𝐜B|PD (6-15)

with ��P|PD being the actual OBM measurement of the MAM position relative to the detector, 𝐜B|P

D

being the position of the detector in SC body frame and δ𝛈OBM,pos =

[δ𝜂OBM,pos,x δ𝜂OBM,pos,y δ𝜂OBM,pos,z]T being a vector of small measurement errors on the position.

Third, the OBM measurement of the MAM platform orientation:

��BP = ��D

P 𝐑BD = 𝛿𝐑OBM, 𝐑D

P 𝐑BD ≈ (𝕀3x3 − [δ𝛈OBM,rot]x) 𝐑B

P (6-16)

with δ𝛈OBM,rot = [δ𝜂OBM,rot,x δ𝜂OBM,rot,y δ𝜂OBM,rot,z]T being a vector of small measurement

errors around the {P}-frame axes.

With these measurements, the knowledge errors on the previously defined quantities of interest are determined as follows.

MOA to LoS misalignment

The MOA misalignment is computed from the measured MAM orientation and the LoS orientation reconstructed from the measured MAM position. Thus, the knowledge error includes both, OBM position and orientation measurement error:

��PLoS 𝐑P

TLoS = ��BLoS⏟ Reconstr.from OBMpos. meas.

��BTP

⏟OBMorient.meas.

𝐑PTLoS

(6-17)

= ��DLoS 𝐑B

D ��BTP 𝐑P

TLoS

=

(

𝕀3x3 − 𝐑P

LoS [δ𝛈OBM,rot]𝐱T𝐑LoSP

⏟ OBM orient. meas. error

+1

‖ ��PD ‖

[

δ𝜂OBM,pos,yδ𝜂OBM,pos,x

0

]

x

𝐑LoSD

⏟ OBM pos. meas. error )

SC attitude in inertial space

The SC attitude is reconstructed from the STR measurement and the OBM orientation measurement. The knowledge error thus includes the STR measurement error as well as the OBM orientation measurement error:


��IB 𝐑I

TB = ��BTP

⏟OBMorient.meas.

��IP⏟STR

𝐑ITB

(6-18)

= ((𝕀3x3 − [δ𝛈OBM,rot]x) 𝐑B

P )T

((𝕀3x3 − [δ𝛈STR]x) 𝐑IP ) 𝐑I

TB

=

(

𝕀3x3 − 𝐑P

B [δ𝛈STR]x 𝐑BP

⏟ STR noise added in

{P}−frame

− 𝐑PB [δ𝛈OBM,rot]𝐱

T𝐑BP

⏟ OBM rot. meas. noiseadded in {P}−frame )

LoS orientation in inertial space

Without the OBM measurement taken into account, the STR measurement is directly used as measurement of the LoS orientation in inertial space. As before, the knowledge error then includes the STR measurement error as well as the MOA misalignment:

��ILoS 𝐑I

TLoS = ��IP⏟STR

𝐑ITLoS

(6-19)

= (𝕀3x3 − [δ𝛈STR]x) 𝐑IP 𝐑LoS

I

=

(

𝕀3x3 − [δ𝛈STR]x⏟ STR noise added in

{P}−frame )

𝐑LoSP⏟ MOA

misalign.

With the MOA misalignment computed from the OBM measurements taken into account, the knowledge error of the LoS orientation in inertial space does include the STR measurement error as well as the OBM position and orientation measurement errors but not the MOA misalignment anymore:

��ILoS 𝐑I

TLoS =

(

��PLoS⏟ Reconstr.from OBMmeas.

��IP⏟STR

)

𝐑ITLoS

(6-20)

=

(

𝕀3x3 − 𝐑P

LoS [δ𝛈OBM,rot]𝐱T𝐑LoSP

⏟ OBM orient. meas. error

− 𝐑PLoS [δ𝛈STR]x 𝐑LoS

P⏟

STR meas. error

+

1

‖ ��PD ‖

[

δ𝜂OBM,pos,yδ𝜂OBM,pos,x

0

]

x

𝐑LoSD

⏟ OBM pos. meas. error )


6.3.3 On-Board Metrology

The position and orientation of the MAM relative to the SC body can be measured by a camera mounted on the MAM facing towards the detectors and a set of IR LED beacons around each of the instruments. Based on the known 3D positions of the beacons and the known mapping onto 2D points on the image sensor by a calibrated camera, the position and orientation of the MAM relative to the SC can be computed. This approach is known as 3D pose estimation. A comparison of iterative algorithms to solve for the position and orientation of a body based on a 2D projection is provided in [30]. To be included in the simulations, a simplified model of this form of OBM represents the measurement errors on position and orientation of the MAM relative to the SC by band-limited white noise with no correlation between the axes. However, correlation should be further analyzed in the future and a more detailed model is required. Additionally, a time-delay can be include to account for the real-time computations of the 3D pose based on a 2D projection.

6.4 Comparison

Table 6-1 lists the effects included in the knowledge errors for both approaches, MOA misalignment measurement and MAM absolute pose measurement. This comparison and the related formulas for the knowledge errors allow a justified design trade-off based on the knowledge requirements and the expected accuracy of the different measurement techniques.


Table 6-1: Comparison of different state determination approaches in terms of knowledge errors

Knowledge Errors

LoS Inertial Attitude SC Inertial Attitude MOA Misalignment

MOA Misalignment Meas. Based Approach

Without OBM meas.:

• STR error

• MOA misalignment

• STR error

• HSM error

• Thermal Distortions

• OBM error

With OBM meas.:

• STR error

• OBM error

MAM Absolute Pose Meas. Based Approach

Without OBM meas.:

• STR error

• MOA misalignment

• STR error

• OBM rot. error

• OBM pos. error

• OBM rot. error

With OBM meas.:

• STR error

• OBM pos. error

• OBM rot. error

In addition to the performance of the different approaches, also the technology readiness level needs to be considered. Table 6-2 provides a classification of the technology readiness level of the different approaches on a scale from -- to ++, where ++ indicates a technology that has already been flown on European missions, + indicates a technology that has been flown with a different use-case on European missions and requires minor modifications, - indicates a technology that uses components that have been flown on a European mission before but in a different use-case and requiring major modifications e.g. in software but not hardware and -- indicates a technology that requires new hardware development.

Table 6-2: Comparison of different state determination approaches in terms of technology readiness level

Technology Readiness Level

STR OBM HSM

MOA Misalignment Meas. Based Approach ++ -

Step Count: +

Optical Encoders: --

MAM Absolute Pose Meas. Based Approach

++ - N.A.

Based on this assessment the MOA misalignment measurement based approach with hexapod state determination via step count has been chosen and implemented for the case study within this thesis.

75

7 Maneuver Guidance with Hexapod in the

Loop Maneuver Guidance with Hexapod in the Loop

7.1 Overview

The guidance for a maneuver between two observations includes two components: The hexapod maneuver and the SC slew maneuver. The hexapod maneuver includes the following elements or a subset of them:

• Instrument switch: +3.7/-2.2° rotation around the y-axis.

The MAM needs to be rotated towards the detector of the selected instrument if an instrument switch is to be performed before the next observation.

• Focus adjustment: +15/-5 mm displacement along the LoS.

The MAM distance from the detector along the LoS needs to be adjusted to set the correct focus length.

• Corrective maneuver: ±2 arcsec rotation around x- and y-axes.

The misalignment between the mirror optical axis and the LoS due to thermal distortions needs to be corrected by a small rotation of the MAM to align the MOA with the LoS.

Additionally, the option exists to perform so called dithering maneuvers with the hexapod during an observation. These dithering maneuvers are Lissajous or raster pointing within a region of 30x30 arcsec to move the image of the observation target across the detector in a certain pattern. This is done to measure or compensate variations in pixel sensitivity. This can be done by slewing the whole SC. However, it can also be possible to perform these maneuvers by moving the MAM along the x- and y-axes and thereby move the LoS across the detector module instead of slewing the whole SC. Note that this option has not been analyzed further in the scope of this thesis.

Rotate MAM

Move MAM

Rotate MAM

Move MAM

Instrument Switch Focus Corrective Manoeuvres Dithering (optional)

Line of Sight Mirror Optical AxisLine of Sight Mirror Optical Axis

Figure 7-1: Hexapod maneuver overview

With the SC slew maneuver, the SC needs to be slewed towards the new attitude between two observations such that the LoS after the instrument switch points towards the new target. In Chapter 7.2, two different operational flows are discussed, defining in which order these maneuvers

76 7. Maneuver Guidance with Hexapod in the Loop

are performed. Thereafter, the LoS guidance with a hexapod in the loop based on the input parameters for a maneuver provided via telecommands is derived in Chapter 7.3. Finally, trajectory generation algorithms are derived for both, the SC in Chapter 7.4 and the hexapod in Chapter 7.5.

7.2 Operational Flow

The above described maneuvers could potentially be performed in a variety of different chronological orders, referred to as ‘operational flow’ hereafter. Two different operational flows have been discussed: Nominal Operational Flow (NOF) with subsequent hexapod and SC maneuvers and Enhanced Operational Flow (EOF) with parallel hexapod and SC maneuvers.

The nominal operational flow is the current baseline suggested by ESA. As mentioned before, the hexapod and SC maneuvers are performed subsequently: First, the instrument switch and focus adjustments are performed with the hexapod. These can be performed in one maneuver, driving the MAM towards the desired position and orientation. Second, the corrective maneuver is performed to align the LoS with the MOA to compensate the thermal distortions of the telescope structure after an OBM measurement. Third, the whole SC is slewed towards the new target line. The advantage of this method is that the SC attitude controller can be in idle mode during the hexapod maneuver. Thus, the attitude controller is not affected by any disturbances caused by the hexapod motion. The inertia uncertainty due to variations in the MAM position and orientation are time-constant seen from the perspective of the attitude controller. The disadvantage is that the subsequent execution of the maneuvers leads to a longer transition time between two maneuvers. In order to fulfill the operational availability requirement, the hexapod maneuver must be performed in a relatively short amount of time (<600s). This results in high actuator speed requirements. The actuator design however requires a trade-off between actuator speed on the one hand and positioning accuracy on the other. However, the positioning accuracy requirements for the hexapod are demanding as well.

The enhanced operational flow is a possible improvement to the baseline. As mentioned before, the hexapod and SC maneuvers are performed parallel in time: The SC is slewed towards the desired new attitude and at the same time the hexapod maneuvers are performed in the same order as before. The advantages of this method are a potentially reduced total transition time as well as potentially more time for the hexapod maneuvers and thus reduced actuator speed requirements. The disadvantage is a more complicated SC attitude controller design due to time-variant parameters (moment of inertia) and disturbances caused by the hexapod movements during the slew. Additionally, the SC attitude guidance and control feedback are complicated by the fact that the STR is mounted on the MAM, which is now moving and rotating during the SC slew maneuver.

The advantages and disadvantages of the nominal and enhanced operational flow are further analyzed and discussed in the reference study in Chapter 8. They are also depicted for comparison in Figure 7-2. In this illustration, the three previously mentioned key differences between NOF on the left and EOF on the right can be observed: First, in NOF the attitude control system (ACS) is in idle mode during the hexapod maneuvers and thus not influenced by the MAM movements. Second, the total transition time of the EOF is shorter because the hexapod maneuvers are performed during the slew. Third, the hexapod maneuvers are performed slower in EOF, because they are executed in the same time as the usually longer SC target slew.

7. Maneuver Guidance with Hexapod in the Loop 77

SC

ACS Mode

timetime

Target Slew

idle slew

Target Slew

slew steady state

(a) Nominal Operational Flow (b) Enhanced Operational Flow

Hex.Instrument

Switch +Focus

LoS Corr.

Instrument Switch +

Focus

LoSCorr.

Figure 7-2: Comparison of (a) nominal and (b) enhanced operational flow

The above stated potential to reduce the actuator speed requirements and the total transition time need to be analyzed in more detail as explained hereafter. The time 𝑡Hex allocated for the hexapod maneuvers is the driver for the actuator speed requirements. As mentioned before, for NOF this time has been set to a maximum of 600 sec. The EOF could relax this requirement to the same time 𝑡S that is required for the SC slew. For large angle slew, 𝑡S is significantly longer than 600 sec and thus more time is available for the hexapod maneuver and thus the actuator speed requirements could be relaxed. However, once the actuator speed requirements are relaxed, the minimum time required for the hexapod maneuvers increases. Thus, for small angle slew, and therefore short slew time 𝑡S the hexapod maneuver time 𝑡Hex can become the driver for the total transition time 𝑡 between two observations as exemplarily illustrated in Figure 7-3. Therefore, further analysis is required to find the optimal actuator speed requirement that maximizes total availability over the complete mock observation plan, cf. Chapter 1.2.1.

Nominal Operational Flow:(subsequent slew)

Enhanced Operational Flow:(simultaneous slew)

hex. man. slew maneuver observation hex. man. slew

hex. maneuver

slew maneuverobservation

slew

hex. maneuver

tHex,NOFtHex,NOF tS1tS1

t1,NOF = tHex,NOF + tS1t1,NOF = tHex,NOF + tS1

tHex,NOFtHex,NOF tS2tS2

t2,NOF = tHex,NOF + tS2t2,NOF = tHex,NOF + tS2

t1,EOF = tS1t1,EOF = tS1 tS2tS2

t2,EOF = tHex,EOFt2,EOF = tHex,EOF

t1,EOF < t1,NOF

t2,EOF > t2,NOF

Figure 7-3: Maneuver time for nominal and enhanced operational flow

Table 7-1 provides a summary of the comparison between NOF and EOF above.


Table 7-1: Comparison of nominal and enhanced operational flow

Nominal Operational Flow (Baseline) Enhanced Operational Flow

Description

Subsequent ISM and SC maneuvers. AOCS in idle mode during ISM maneuver.

Simultaneous ISM and SC maneuvers.

Comparison

Advantages • Decoupled AOCS and ISM controller. • Potential to increase availability of the SC due to shorter transitions between two observations.

• Potential of reduced hardware requirements.

• Potential to use sequential actuator control concept with fewer hardware components needed.

Disadvantages • Sequential actuator control concept not possible.

• Higher robustness requirements for AOCS design.

• Challenges in stability analysis due to time-variant MoI and disturbance torques caused by moving hexapod.

7.3 Line of Sight Guidance with Hexapod in the Loop

In the following the SC and hexapod guidance algorithms are derived for both operational flows. The inputs for the guidance generation are the desired LoS attitude in inertial space for the next observation, the desired instrument for the next observation and the desired focus distance for the instrument. These inputs are provided via telecommands or are stored as parameters on board. Based on these inputs, the guidance trajectories for the hexapod and the SC attitude control system are computed as follows: The required hexapod pose is defined by the instrument selection, the required focus distance and the thermal distortions of the telescope structure that are measured by the OBM and need to be compensated with the hexapod. Given these inputs, the required position and orientation of the hexapod platform can be computed and a trajectory can be generated under given actuator constraints to drive the hexapod towards its new target pose from the current pose as described in Chapter 7.5. The required SC attitude is computed based on the required target orientation of the LoS and the previously computed orientation of the MAM relative to the SC at the end of the hexapod maneuver. A SC attitude trajectory is then computed to slew the SC from its current attitude towards its desired new attitude as described in Chapter 7.4. Figure 7-4 further illustrates the relation between LoS and SC attitude: On the left side, the configuration before the repointing maneuver is shown, with the hexapod still being in its previous pose from the last observations. On the right side, the final configuration after the hexapod and the slew maneuver is shown, with the hexapod being in its new pose and the SC being at its new attitude such that the LoS and the target line are aligned. As can be seen, the required attitude change is not equal to the required LoS change due to the different hexapod poses before and after the maneuver.


old target old target

new target

LoS change SC attitude change

Line of Sight SC attitude

Figure 7-4: SC attitude change during repointing maneuver

7.4 Spacecraft Attitude Trajectory

Two approaches have been discussed for the SC slew maneuver between two observations without violating the sun exclusion zone, which are called Classical Re-Pointing approach (CRP) and Enhanced Re-Pointing approach (ERP) hereafter. Figure 7-5 illustrates the trajectories in terms of the SC telescope boresight trace from an initial boresight direction BD1 to a final boresight direction BD2 for CRP in (a) and ERP in (b). The SEZ half-cone is shown in grey around the sun vector along the x-axis. Note that the term boresight is used here to refer to the direction of the SC tube, i.e. the negative zB-axis of the SC body frame, not the LoS. Figure 7-6 illustrates the azimuth and elevation angles as well as the rotation angle around the zB-axis over time for both re-pointing approaches as before.

(a) (b)

Figure 7-5: SC boresight trace for (a) classical and (b) enhanced re-pointing approach shown in {I}-frame

(a) (b)

Figure 7-6: Rotation angles over time for (a) classical and (b) enhanced re-pointing approach

BD1

BD2

BD1

BD2

BDI


The classical re-pointing approach is heritage from previous Airbus missions and can be used as a solid baseline. It avoids the SEZ by: First, a rotation around the yB-axis of the SC body frame (elevation) bringing the SC boresight from BD1 back into the azimuth plane, i.e. the y-z-plane in Figure 7-5. Second, a rotation around the xB-axis (azimuth), leading to a rotation of the SC boresight within the azimuth plane. Third, another rotation around the yB-axis bringing the SC boresight out of the azimuth plane to the new elevation angle at the final boresight direction BD2.

The enhanced re-pointing approach has been derived by Hablani in [31]. It avoids the SEZ by: First, a rotation around the xB-axis to an intermediate boresight direction BDI. Second, a rotation around the zB-axis at this intermediate point. Third, another rotation around the xB-axis to the final boresight direction BD2. This approach is more time-efficient due to the shorter path, but requires further technology development and analysis as shown in the comparison in Table 7-2. Note that the ERP includes a rotation around the zB-axis. Depending on the design of the SC this may lead to stray-light falling into the telescope during the slew maneuver, e.g. due to a sun-shield design as shown in Figure 1-4, which provides protection against stray-light only for a certain area. Stray-light falling into the telescope during observations leads to the immediate destruction of the instruments and thus mission failure. Whether this is also the case during non-operations periods needs to be analyzed further.

Both approaches include three sequential single axis rotations (Y-X-Y for classical re-pointing approach and X-Z-X for enhanced re-pointing approach). A bang-slew-bang trajectory is generated for each of these rotations as described in Appendix 0. Such a trajectory is defined by three phases. First, during acceleration phase, the SC is accelerated with a constant acceleration rate up to a maximum angular velocity. Second, during constant speed phase, the SC slews with a constant angular velocity for a certain time. Third, during deceleration phase, the SC is brought to a stop with the same but negative acceleration rate as before. Figure 7-7 exemplarily illustrates such a maneuver, showing angular acceleration, velocity and rotation angle over time. Note that depending on the required rotation angle, the maximum velocity may not be reached as illustrated in Figure 7-7 (a). In this case, the deceleration phase follows right after the acceleration phase, also often referred to as bang-bang maneuver.

(a)

t

ω

t

ω

t

ω

t

ϴ

t

ω

t

ω

t

ϴ

(b)

t

ω

t

ω

t

ϴ

Figure 7-7: Exemplary illustration of a bang-bang maneuver in (a) and bang-slew-bang maneuver in (b)

Table 7-2 provides a comparison of the classical and enhanced repointing approach as given in [1] on a scale from -- for worst to ++ for best and TBC indicating a preliminary classification with need for further analysis.


Table 7-2: Re-pointing design trade-off considerations

Trade-Off Consideration Classical Re-Pointing Enhanced Re-Pointing

CPU resource needs and computation time

++ - (TBC)

Design complexity introduced by the selected slew implementation

++ + (TBC)

Re-pointing maneuver time - +

Operational cost + +

Another point that needs to be analyzed further is the change in thermal distortions of the telescope due to the attitude changes during and after the slew maneuver. In general, a change in intensity of the sun radiation onto the surface leads to expansion or shortening of the telescope structure. A change in intensity can be due to two causes: First, a change of the irradiation angle and, second, a change from shadow to sun exposure or the other way around. In CRP, changes in irradiation angle are the main cause for the changes in thermal distortions of the telescope. The largest changes occur in bending around the yB-axis due to elevation changes. The largest changes are between 0° elevation (sun irradiation angle 90° to surface) and the maximum elevation angle of 30° (sun irradiation angle 60° to surface). In CRP, the SC is always rotated back into the azimuth plane, i.e. 0° elevation. This leads to changes in distortions even if the initial and final elevation angles are the same. In ERP elevation changes may be smaller than in CRP since the SC is not rotated back into the azimuth plane if there is no sign change in elevation angle. However, the rotation around the zB-axis potentially leads to relevant changes in the thermal distortions, because parts of the telescope structure that have previously been in the shadow then become exposed to the sun and the other way around. Therefore, thermal distortions need to be further analyzed and compared for both approaches.

7.5 Hexapod Pose Trajectory

Two approaches have been discussed for the hexapod maneuver between two observations without violating the actuator constraints, which are called hexapod state domain guidance and hexapod actuator domain guidance hereafter. In hexapod state domain guidance, a trajectory with constant acceleration and deceleration like the one described in the previous chapter is generated to drive each hexapod state from its initial value to its final value. In hexapod actuator state domain guidance, a trajectory with constant acceleration and deceleration is generated to drive each actuator from its initial length to its final length. Note that with the actuators being only open-loop controlled, it must be ensured in both approaches, that the given trajectory does not violate any actuator limitations, meaning that the actuators have enough actuation capacity to follow the trajectory without any step loss or other faults. Both approaches are discussed hereafter in more detail.

7.5.1 Hexapod Actuator Domain Guidance Algorithm

In actuator state domain guidance, a trajectory is generated using the following algorithm, which is also depicted in a flow chart in Figure 7-8. First, the current actuator lengths are computed according to the initial hexapod pose using the hexapod inverse kinematic. The desired actuator lengths are computed accordingly for the given target pose and the distance each actuator needs to travel from the current to the target pose is computed. Second, the maneuver time 𝑡m is computed based on the actuator with the largest travel distance under consideration of the actuator limitations. Third, a bang-slew-bang like trajectory is generated for every actuator to reach its end point within 𝑡m as described in Appendix 0.


Maneuver time computationHexapod target pose

Inv. Kin.

Hexapod current

pose

Inv. Kin.

Target actuator length

Current actuator length

+-

ΔlCompute maneuver

time

tman

BSB trajectory generation

Actuator trajectory

Figure 7-8: Hexapod actuator state domain guidance algorithm flow chart

Figure 7-9 exemplarily shows the actuator lengths over time on the left and the hexapod states over time on the right. As can be seen, the actuator lengths are monotonously increasing according to the bang-slew-bang like trajectory. Due to the non-linearity of the hexapod kinematic, this does not lead to a monotonous change in hexapod states. Instead it can be seen that some states first change in the opposite direction in this example.

(a) (b)

Figure 7-9: Exemplary plot of (a) actuator lengths and (b) hexapod states over time for actuator state domain guidance

The advantages of the actuator state domain guidance algorithm are a simple implementation and easy consideration of actuator limitations such as maximum velocity and acceleration as well as existing algorithms to generate the actual step pulses for the motors to follow the guidance profile, cf. Chapter 7.5.4. The disadvantage is that the hexapod platform follows a ‘non-smooth’ path to the end position due to the non-linearity of the hexapod kinematic. This affects the disturbance torque caused by the moving MAM mass and thus also the SC attitude control. However, these effects are relatively small and could also be compensated through feedforward of the hexapod position during the slew if necessary.

7.5.2 Hexapod State Domain Guidance Algorithm

In hexapod state domain guidance, a trajectory is generated using the following algorithm, which is also depicted in a flow chart in Figure 7-10. First, the maneuver time 𝑡m is computed as described in the previous subchapter. Second, the difference in hexapod states between initial and target pose is computed. Third, a bang-slew-bang like trajectory is generated for each hexapod state to reach its end


value within 𝑡m as described in Annex 0 and converted into actuator trajectories using the inverse kinematic of the hexapod. Fourth, the actuator trajectories are checked for violations of the actuator limitations (maximum speed and velocity). If a violation occurs, the maneuver time is increased by a predefined increment and the last two steps are repeated.

Maneuver time

comp.

Hexapod target pose

Hexapod current

pose

Compute desired

hexapod state

change

tman

ΔxHE

BSB trajectory generation

ΔtmanΔtman+

Hex. state traj.

Hex. state traj.

Act. length

traj.

Act. length

traj.

Act. length

traj.

Act. length

traj.

Maneuver time

adjust.

Inv. Kin.

act. constr. violation?

yes

no

Figure 7-10: Hexapod state domain guidance algorithm flow chart

Figure 7-11 exemplarily shows the actuator lengths over time on the left and the hexapod states over time on the right. As can be seen, the hexapod states are now monotonously increasing according to the bang-slew-bang like trajectory. Therefore, due to the non-linearity of the hexapod kinematic, the actuator lengths now follow a ‘non-smooth’ trajectory. To visualize this better, Figure 7-12 shows the difference between the actuator lengths from the actuator state domain guidance given in Figure 7-9 (a) and the ones from hexapod state domain guidance shown here.

(a) (b)

Figure 7-11: Exemplary plot of (a) actuator lengths and (b) hexapod states over time for hexapod state domain guidance


Figure 7-12: Difference between actuator lengths for hexapod state domain guidance compared to actuator state domain guidance

The advantages of the hexapod state domain guidance algorithm is a smooth transition from one pose to another with monotonous change in position and orientation values. The disadvantages are that it is more complicated to respect the actuator limitations in the algorithm and to generate the step pulses for the motors for the ‘non-smooth’ actuator trajectory.

7.5.3 Comparison

It has been decided to use the actuator state domain guidance here, based on the following two trade-offs: First, the actual movement of the platform during the maneuver is only of relevance for dithering maneuvers during an observation (cf. Chapter 7.1), which are not considered within this thesis and can be done by rotating the whole SC instead. Second, the non-linear actuator speed profiles resulting from the hexapod kinematic in the hexapod state domain guidance complicate the actual step command generation for the actuators while for the linear-speed ramps resulting from the actuator state domain guidance well proven simple algorithms exist as described in the next chapter.

7.5.4 Step Command Generation

For a given linear speed profile as derived in Chapter 7.5.1, Austin proposed a simple algorithm in [32] for the timing of the discrete motor step commands. This algorithm can not only be run on low-end hardware and uses only simple fixed-point arithmetic operations, it also does not use extensive data tables as preceding approaches did [32].

In general, a faster interval of step commands leads to a higher motor speed. Thus, the motor steps need to be timed such that the motor follows the linear speed ramp derived in Chapter 7.5.1. The motors are controlled with a certain frequency 𝑓. This means that every 𝑓−1 sec it needs to be decided whether to perform a step or not. Therefore, a timer with the same frequency is used here and a counter that increments by one every time it is triggered by the timer. Then, a given linear speed profile needs to be converted into a sequence of timer counts, specifying each time a step needs to be performed. From the given travel distance Δ𝑙, maximum velocity 𝑣max during the constant speed phase of the maneuver and acceleration 𝑎 of the linear actuator, all derived in Appendix 0, the total number of steps 𝑛tot, the maximum angular velocity 𝜔max during the constant speed phase and the angular acceleration �� of the motor shaft can be computed:


𝑛tot = round (|Δ𝑙|

𝛼s)

(7-1)

𝜔max = 2𝜋𝑣max

𝛼m

(7-2)

�� = 2𝜋𝑎

𝛼m

(7-3)

where 𝛼s is the actuator stroke per step, 𝛼m is the actuator stroke per motor revision and round(∙) returns the nearest integer. For a constant motor acceleration, the angular velocity is a linear function of time:

𝜔(𝑡) = ��𝑡 (7-4)

The shaft angle 𝜃 at time 𝑡 is then defined as:

𝜃(𝑡) = ∫ 𝜔(𝜏)𝑑𝜏𝑡

0 = 1

2𝜔𝑡2

(7-5) = 𝑛𝛼s

with 𝑛 being the number of steps performed up to time 𝑡. Solving Eq. (7-5) for 𝑡 leads to the time when the nth step needs to be performed:

𝑡n = √2𝑛𝛼s

��

(7-6)

The timer count 𝑐n from step 𝑛 to 𝑛+1 is then computed as:

𝑐n = (𝑡n+1 − 𝑡n)𝑓 = (√

2(𝑛+1)𝛼s

��− √

2𝑛𝛼s

��)𝑓

(7-7)

= (√𝑛 + 1 − √𝑛)√

2𝛼s

��𝑓

With 𝑐0 denoting the timer count between the maneuver start and the initial step, Eq. (7-7) can be rewritten as:

𝑐n = (√𝑛 + 1 − √𝑛)𝑐0 (7-8)

Where the initial timer count is computed from Eq. (7-7):

𝑐0 = √2𝛼s

��𝑓

(7-9)

The timer count between two steps during the constant speed phase can be computed from 𝜔max as:

𝑐𝜔max = 2𝜋𝑓

𝜂𝜔max

(7-10)

with 𝜂 being the number of steps per motor revision. The number of steps required to accelerate towards the final velocity is computed by inserting Eq. (7-10) into Eq. (7-8) and solving for 𝑛:


𝑛acc = round (

(𝑐02−𝑐𝜔max

2 )2

4𝑐02𝑐𝜔max2 )

(7-11)

The number of steps required to deaccelerate at the end maneuver is the same and thus the number of steps required to be performed during the constant speed phase is given by:

𝑛𝜔max = 𝑛tot − 2𝑛acc (7-12)

To assemble the complete time count sequence, the timer counts for steps 1…𝑛acc are computed using Eq. (7-8), then 𝑐𝜔max is attached at the end of the sequence 𝑛𝜔max times and finally the inverted

acceleration sequence is attached for deceleration. Note that [32] also suggest an approximation for Eq. (7-8) by a Taylor series expansion of the square roots to further decrease the computational burden of this algorithm. For the given application however, the timer count sequence does not need to be computed in real time but can be computed a priori for a whole maneuver. The exact solution including the square roots has thus been implemented here.

Note that to compensate the hysteresis effect of the actuator gear box and spindle described in Chapter 4.4, an additional number of steps needs to be added according to the deadband width whenever an actuator changes direction. However, these steps cannot simply be added to the total number of steps 𝑛tot. Instead, the spindle needs to be driven towards the opposite contact point in the thread slowly. Thus, a second maneuver with constant acceleration and deceleration as described above needs to be performed upfront the actual maneuver with every actuator that changes direction compared to its last movement. Note that during these hysteresis compensation maneuvers the piston of the linear actuators do not move even though the motor moves. Figure 7-13 further illustrates this. The left plot shows the commanded trajectory of one actuator over time as derived in Chapter 7.5.1. The right plot shows the commanded actuator steps vs actual actuator length.

(a) (b)

Figure 7-13:Executed motor steps in (a) and commanded actuator length vs actual actuator length in (b)

hysteresis compensation

87

8 Reference Case Study

Reference Case Study

8.1 Overview

In this chapter, the case study and corresponding parameters and results are discussed. First, the parametrization of the spacecraft as well as characteristics of the thermal distortions and the parametrization of the hexapod are provided in Chapter 8.2. Furthermore, the re-pointing scenario simulated in this case study is defined. Second, the hexapod induced disturbances are discussed and characterized for the given reference spacecraft and scenario in Chapter 8.3 as well as the time-variant inertia uncertainty due to the moving MAM mass. Third, different simulation cases are defined in Chapter 8.4 and finally compared in terms of performance results in Chapter 8.5. The design trade-offs based on these results are then discussed in Chapter 9.

8.2 Parametrization

8.2.1 Reference Spacecraft Parameters

This chapter provides the parameters for the reference spacecraft that has been used in this case study and is based on but not identical to the current ESA design concept. Table 8-1 provides the mass and inertia properties of the SC and the MAM. Table 8-2 provides all relevant dimensions of the SC and the hexapod mechanism as defined in Chapter 3.4.

Table 8-1: SC and MAM mass and inertia properties

Mass [kg]

SC without MAM 𝑚S 6000 kg

MAM 𝑚P 2000 kg

Moment of Inertia [kg*m2]

SC MoI wrt. SC CoG 𝓘S(rot)|S [190000 50 200050 180000 −1002000 −100 30000

] kgm2

MAM MoI wrt. MAM CoG 𝓘P(rot)|P [2500 0 100 2500 020 0 3000

] kg m2

88 8. Reference Case Study

Table 8-2: SC and hexapod dimensions

Detector Positions [mm]

X-IFU position relative to SC body frame 𝐜B D1|B [−500 0 6000]|BT mm

WFI large position relative to SC body frame 𝐜B D2|B [600 0 6000]|BT mm

WFI fast position relative to SC body frame 𝐜B D3|B [750 0 6000]|BT mm

Hexapod Mechanism Position and Dimensions [mm], [deg]

Hexapod base position relative to SC body frame

𝐜B H|B [0 0 −5500]|BT mm

Nominal MAM position relative to hexapod base

𝐜H P0|H [0 0 500]|BT mm

Hexapod base junction point positions (cf. Figure 2-6)

𝛾H 26°

𝛾P 0°

𝑟H 1300 mm

𝑟P 1200 mm

8.2.2 Re-Pointing Scenario Parameters

Table 8-3 provides the SC and hexapod configuration before and after the re-pointing maneuver.

Table 8-3: SC and hexapod configuration before and after re-pointing maneuver

LoS Orientation in Inertial Space [deg]

Parameter Initial Final

Elevation angle 38.7° 32.8°

Azimuth angle 0° 120°

ISM Configuration

Detector selection WFI-Fast X-IFU

Focus distance 12 m 12 m + 15 mm

Note that because the LoS orientation relative to the SC changes when another detector is selected, the SC elevation angle is 35° in both cases. This can be seen in Figure 8-1 depicting the initial and final configurations.

8. Reference Case Study 89

(a) WFI (b) X-IFU

Figure 8-1: SC attitude and ISM configuration (a) before and (b) after re-pointing maneuver

Table 8-4 provides the maneuver start times and durations for both, NOF and EOF.

Table 8-4: Operational flow maneuver start times and duration

Hexapod Maneuver

Parameter NOF EOF

Maneuver start time 100 sec 100 sec

Maneuver duration 600 sec 7000 sec

SC Slew Maneuver

Maneuver start time 1300 sec 100 sec

8.3 Characterization of Hexapod Effects on Spacecraft Attitude Control

The effects of the moving hexapod on the SC attitude control have been quantified for the given parametrization. The hexapod induced disturbance torque is analyzed in Chapter 8.3.1 and the time-variant inertia uncertainties are analyzed in Chapter 8.3.2.

8.3.1 Hexapod Induced Disturbance Torque

The disturbances caused by the moving MAM mass without considering the quantization due to the stepper motors is shown below in Figure 8-2 for the nominal and enhanced operational flow.

(a) (b)

Figure 8-2: Hexapod induced disturbances over time WITHOUT step quantization for (a) NOF and (b) EOF

35° SC

elevation

35° SC

elevation

120° SC

azimuth

Nominal Operational Flow Enhanced Operational Flow


The disturbances caused by the moving MAM mass with the stepper motor quantization however include additional high amplitude noise. Figure 8-3 shows the square root of the cumulated power spectrum of the disturbance. As can be seen, most of the power of these disturbances is in higher frequencies, not affecting the attitude controller which has a cut-off frequency at 1Hz. However, there is still a relevant contribution (<0.5mNm up to 1Hz) within the ACS bandwidth that causes problems for the enhanced operation flow. In the controller design a certain amount of the reaction wheel actuation capacity is allocated for feed forward on the one hand and model following on the other hand. The feed forward torque allocation is used to follow the given guidance trajectory and the model follow torque allocation to react to control errors due to model uncertainties and disturbances. With the current allocation, the step quantization induced torque noise is too high to be attenuated by the controller and thus it is not able to follow the guidance trajectory anymore. Note that for the nominal operation flow this is irrelevant as the ACS is deactivated during the hexapod maneuver. Therefore, the ACS only compensates the attitude error caused by the hexapod maneuver after it has finished. In the enhanced operational flow, the ACS is active during the hexapod maneuver and thus needs to compensate the hexapod induced disturbances within its bandwidth.

Figure 8-3: Square root of cumulated disturbance power spectrum WITH step quantization

The following two options exist to enable the EOF: First, increase the actuation capacity allocated for disturbance attenuation. Second, decrease the power of the disturbances within the ACS bandwidth. Since maneuver time between two observations is a critical factor for the Athena mission, the actuation capacity allocated for disturbance attenuation shall not be further increase, because this leaves less capacity to perform the actual slew maneuver and thus increases the slew maneuver time. Instead, it has been attempted to shift the disturbances into higher frequencies by changing the actuator design to perform smaller steps with higher frequency. The force that is required to accelerate

the MAM platform is proportional to the acceleration 𝑎 = 2 ∗ 𝑠step/𝑡step2 . To keep the design and

dimensioning of the actuators the same, the acceleration during a step shall remain the same. Thus, when the step frequency is for example increased by a factor of 2, the stroke per step length must be decreased by a factor of 4 to keep the acceleration the same. This decreases the actuators’ maximum speed by a factor of 2 while the step frequency is 2 times higher and thus shifts the spectrum of the step induced disturbance towards higher frequencies. Note that the decreased stroke per step ratio also increases the positioning accuracy of the hexapod while the decreased maximum velocity is not necessarily a problem in the enhanced operation flow. However, it needs to be further analyzed if this shift causes problems in other disciplines, e.g. micro-vibration analysis.

8. Reference Case Study 91

8.3.2 Time-Variant Inertia Uncertainty

Figure 8-4 shows the inertia error due to the change in position and orientation of the MAM over time for nominal and enhanced operational flow. As can be seen, in nominal operational flow on the left, the error is constant over time during the SC slew maneuver and only changes while the ACS is inactive in the beginning. In contrast, the inertia changes also during the slew maneuver in enhanced operational flow on the right.

(a) (b)

Figure 8-4: Hexapod induced inertia error for (a) NOF and (b) EOF

8.4 Simulation Cases

Four different cases have been simulated and compared in terms of performance. These four cases represent all possible combinations of the two operational flows discusses in Chapter 7.2, i.e. NOF and EOF, and the two repointing approaches discussed in Chapter 7.4. Table 8-5 provides an overview of these simulation cases and assigns a reference number to each of them.

Table 8-5: Simulation cases overview and reference number

Operational Flow

Nominal Enhanced

Re

-Po

inti

ng

Ap

pro

ach

Cla

ssic

al

Sim. case

1.1

Sim. case

1.2

Enh

ance

d

Sim. case

2.1

Sim. case

2.2

Note that for the current simulations, the attitude controller has been in steady state mode during the hexapod maneuver instead of idle mode as anticipated. This is because the current implementation requires the SC guidance to be computed prior to the simulation. The control error when activating the attitude controller after the hexapod maneuver then leads to long re-settling times. Therefore, it is planned to adjust the implementation such that the SC guidance is computed online during the simulation based on the actual attitude after the hexapod maneuver. These changes are expected to have a very small impact on the overall performance assessment since the SC attitude changes due to the hexapod maneuver are very small compared to the actual slew. However, major changes to the attitude controller or its implementation have been out of the scope of this thesis.

Nominal Operational Flow Enhanced Operational Flow


8.5 Performance Analysis and Comparison

Table 8-6 compares the different simulation cases in terms of performance (total re-pointing maneuver time) as well as an assessment of the technology readiness of each approach on a scale from -- to ++. The current baseline suggested by ESA is highlighted and the improvements of the other cases is given in percental comparison to the baseline. The time plots for all simulation cases are provided in Appendix B.

Table 8-6: Simulation cases performance comparison

Operational Flow

Nominal Enhanced

Re

-Po

inti

ng

Ap

pro

ach

Cla

ssic

al

Time: 9279 s

(2h 35min)

Time: 8079 s

(2h 15min)

Comp. to BS: N.A. Comp. to BS: 13%

Technology Readiness

++ Technology Readiness

-

Comments: Baseline (BS) Comments: Feasibility with actuator quantization to be shown.

Enh

ance

d

Time: 8015 s

(2h 14min)

Time: 6815 s

(1h 54min)

Comp. to BS: 14% Comp. to BS: 27%

Technology Readiness

- Technology Readiness

--

Comments: Rotation around z-axis. Comments: Rotation around z-axis.

Feasibility with actuator quantization to be shown.

93

9 Pointing System Design Trade-Offs

Pointing System Design Trade-Offs

9.1 Overview

The feasibility of the baseline concept, i.e. nominal operational flow and classical re-pointing approach has been shown in the case study in the previous chapter. However, it has also been mentioned, that the attitude controller implementation needs to be adjusted to compute the trajectory online, based on the current attitude after the hexapod maneuver. Additionally, possible improvements have been analyzed in the case study which require further technology developments as discussed hereafter. The trade-off between the performance improvement of the enhanced operational flow versus the difficulties and required technology developments is discussed in Chapter 9.2. The trade-off between the performance improvement of the enhanced re-pointing approach versus the difficulties and required technology developments is discussed in Chapter 9.3.

9.2 Operational Flow

For the exemplary maneuver simulated in the case study, the enhanced operational flow provided a 13% shorter total maneuver time compared to the baseline. However, to use this approach, further analysis is required in the following two topics. First, the disturbance torque noise caused by the step quantization has been identified as a potential design driver and thus needs to be analyzed further. It has not been included in the simulations presented within this thesis because the current controller is not capable of attenuating these disturbances. Thus, it needs to be evaluated whether the current simplified model is sufficient to represent the effects caused by the step quantization. It then needs to be analyzed further, if the disturbance can be shifted into higher frequencies out of the ACS bandwidth as described in Chapter 8.3.1 and how this would affect other systems of the SC. Second, it needs to be analyzed how a changed actuator design with lower maximum actuator speed would affect the overall availability for observations as discussed in Chapter 7.2.

9.3 Re-Pointing Approach

For the exemplary maneuver simulated in the case study, the enhanced re-pointing approach provided a 14% shorter total maneuver time compared to the baseline. However, to use this approach, the effects of the z-axis rotation included in the maneuver need to be analyzed further as discussed in chapter 7.4. It needs to be analyzed whether sun straylight falling into the telescope during non-observational periods is a problem. If so, the sun-shield may need to be adapted such that it allows z-axis rotations within a required envelope without straylight falling into the telescope. Additionally, it needs to be analyzed how z-axis rotations affect the thermal distortions of the telescope structure.

95

10 Conclusion

Conclusion

10.1 Summary

The three high-level objectives of this thesis, introduced in Chapter 1.3, have been addressed and fulfilled throughout the thesis work as follows.

First, the technology readiness level for a pointing control system with a hexapod in the loop has been increased by closing the gaps in the existing Airbus in-house design and analysis tools. The hexapod open-loop control chain has been analyzed and modelled for simulations including hexapod inverse kinematic and forward kinematic models as well as a new spacecraft attitude dynamic model with prescribed hexapod motion. For the inverse kinematic, an efficient algorithm has been implemented by combining different literature resources. For the forward kinematic, two alternative algorithms have been implemented which can be used for high-accuracy on-ground simulations on the one hand and for less accurate on-board computations on the other. The attitude dynamics have been derived using Newton Euler formulation for a two-body system in free space with prescribed relative motion and the terms for hexapod induced disturbance and inertia variations have been identified in the equations of motion. A simplified actuator model has been developed in close cooperation with the Airbus mechanism department for a first analysis of actuator related effects on the attitude control system. Different pointing system state determination approaches have been compared analytically, providing a set of alternative approaches for future design trade-offs. These approaches have been compared in terms of remaining knowledge errors and technology readiness and one approach has been selected for implementation in the case study. This approach utilizes an on-board metrology which directly measures the misalignment between the line of sight and the mirror optical axis in combination with a software based approach for the hexapod state determination. Algorithms for the complex line of sight guidance with combined spacecraft and hexapod maneuvers have been derived an implemented. All these tools have been summarized in an easy to use Matlab®/Simulink® library for quick integration into existing spacecraft attitude simulations.

Second, different operational scenarios have been designed and put into comparison in closed-loop simulations within a representative case study. A baseline concept, i.e. sequential hexapod and SC maneuvers and a well proven re-pointing approach, has been selected from these scenarios and its feasibility has been shown through simulation. The time-variant parameters and disturbance torque caused by the moving MAM have been characterized through simulation and the disturbance noise caused by the hexapod actuator step quantization has been identified as a potential design driver with the need for further and more detailed analysis. Additionally, the need for online SC slew maneuver guidance computation has been identified as further described in the outlook hereafter.

Third, possible improvements of the other operational scenarios compared to the baseline have been identified and discussed. All operational scenarios have been compared in terms of performance within the case study as well as technology readiness and feasibility. The first improvement is the enhanced operational flow, i.e. performing SC and hexapod maneuvers at the same time. For the exemplary re-pointing scenario simulated in the reference case study, this provided a 13% shorter total transition time between two observations compared to the baseline, i.e. sequential hexapod and SC maneuvers. This approach required the attitude controller to be adapted to account for the additional disturbances caused by the hexapod motion as discussed throughout this thesis. Additionally, the

96 10. Conclusion

analytical stability proof in presence of time variant inertia remains to be done based on the inputs provided by this thesis as discussed in the outlook chapter hereafter. The second improvement is the enhanced re-pointing approach, which provided a 14% shorter total transition time in the reference case study. This approach requires further analysis regarding stray-light potentially falling into the telescope during the slew maneuver and thus a potentially required re-design of the sun-shield structure of the telescope. Combining both improvements led to a 27% shorter transition time for the reference case study and thus may provide a significant increase in availability for science observations.

In conclusion, this thesis showed the feasibility of a relatively simple baseline concept for the Athena line of sight control while also identifying potential design drivers and open design points. It provides a toolset for further analysis with a level of detail that is uncommon for the current status of the project, i.e. a phase A feasibility study. Additionally, it provides a set of potential design improvements and related trade-offs that can be used for future design decisions.

10.2 Outlook

The following five topics require further research and analysis. First, the step quantization related effects have been identified as a potential design driver and thus need to be investigated further. As previously described, it needs to be evaluated whether the currently seen effects of the step quantization truly exist or are only caused by the simplified model that does for example not include the elastic coupling between the gearbox and the spindle of the motor, as shown in Figure 4-8. If these effects are not induced by the simplifications of the model, it needs to be analyzed which design changes can be made to the actuators to solve the problem. For example, adding additional damping elements or shifting the disturbances into higher frequencies by increasing the step frequency of the motors as previously described.

Second, the existing SC guidance algorithms need to be adapted, such that they compute the SC guidance online during the simulation. Currently they need to be run prior to the simulation and can thus not account for the small changes in SC attitude caused by the hexapod maneuvers taking place beforehand in the nominal operational flow. Thus, the precomputed trajectory would be incorrect, requiring unnecessary re-settling periods to return to the initial SC attitude before starting the slew maneuver. This has been temporarily solved by running the SC attitude controller in steady state mode during the hexapod maneuver instead of idle mode as anticipated for nominal operational flow to simplify controller design and stability proof. Note that the hexapod maneuver guidance generation derived and implemented within this thesis is computed online, taking into account the current OBM measurements of the thermal distortions. Thus, this implementation can be used as a framework for the required changes to the existing SC guidance algorithms.

Third, for the enhanced operational flow, stability of the closed-loop needs to be shown analytically. The required inputs are provided within this thesis, i.e. the model uncertainty due to the time-variant inertia. Note that the disturbance torque caused by the hexapod motion has already been accounted for in the robust controller design. Stability in presence of time-variant model uncertainties can for example be shown via the small gain theorem as briefly outlined hereafter. The true system 𝐆 is represented by its nominal transfer function 𝐆0, i.e. with the MAM in its nominal position in this case, and a small deviation 𝚫G such that:

𝐆 = (𝕀3x3 + 𝚫G)𝐆0 (10-1)

Then, the closed-loop with controller 𝐅, as illustrated in Figure 10-1, is stable if 𝐏 is stable, 𝚫G is stable and ‖𝐏‖‖𝚫G‖ < 1. Note that this is a conservative stability check. If it fails, other approaches must be applied for stability analysis.

10. Conclusion 97

F G0

ΔG

GP

Figure 10-1: Hexapod state domain guidance algorithm flow chart

Fourth, a stationary substitute model for the actuator induced disturbance should be derived that includes the relevant actuator effects in hexapod state domain. Thereby the computational burden of computing inverse and forward kinematics in order to model actuator effects in the actuator state domain could be eliminated. This would reduce the simulation time significantly and thus allow simulations over a sequence of several maneuvers or Monte-Carlo simulations to improve the characterization of the hexapod induced disturbance torque.

Finally, an important future research topic is the fault management with the hexapod in the loop on the one hand and autonomous agile maneuvers with the SEZ as a hard constraint whose violation leads to mission loss on the other. The Fault Detection, Isolation and Recovery (FDIR) for the hexapod mechanism, especially with a software-based hexapod state determination approach, is an important and challenging research topic. Additionally, a fault-tolerant ACS design must ensure that the SC never enters the SEZ at any time, also if a fault occurs during an agile slew maneuver with high velocities.

99

Appendix A Derivations and Side Notes

Derivations and Side Notes

A.1 Rotation Matrix Time Derivatives

As derived in [9], the first-order time derivative of a rotation matrix 𝐑12 , transforming a vector 𝐱|1

expressed in {1}-frame to vector 𝐱|2 expressed in {2}-frame, is given by:

��12 = [ 𝛚1 2|2]x

T𝐑12 (A-1)

where 𝛚1 2|2 is the angular velocity of frame {2} relative to frame {1} expressed in frame {2} and[𝐚]x

denotes the skew-symmetric matrix of vector 𝐚 ∈ ℝ3.

[𝐚]x =

[

0 −𝑎z 𝑎y𝑎z 0 −𝑎x−𝑎y 𝑎x 0

]

(A-2)

The second-order time derivative of 𝐑12 , is then computed as:

��12 =

([ ��1 2|2]xT+ ([ 𝛚1 2|2]x

T)2

) 𝐑12

(A-3)

Correspondingly the time derivatives of the rotation matrix 𝐑21 = ( 𝐑1

2 )−1= ( 𝐑1

2 )T

are given as:

��21 = [ 𝛚2 1|1]x

T𝐑21 (A-4)

��21 =

([ ��2 1|1]xT+ ([ 𝛚2 1|1]x

T)2

) 𝐑21

(A-5)

where 𝛚2 1|1 is the angular velocity of frame {1} relative to frame {2} expressed in frame {1} and can

be related to 𝛚1 2|2 as follows:

𝛚2 1|1 = 𝐑21 𝛚2 1|2

(A-6) 𝐑21 (− 𝛚1 2|2)

Alternatively, the time derivatives of the rotation matrix 𝐑12 and its inverse 𝐑2

1 can be related to each other as follows:

��21 = [ ��1

2 ]T

= 𝐑𝟐 1T[ 𝛚1 2|2]x

= 𝐑21 [ 𝛚1 2|2]x

(A-7)

��21 = [ ��1

2 ]T

= 𝐑21 ([ ��1 2|2]x

T+ ([ 𝛚1 2|2]x

T)2

) (A-8)

100 Appendix A: Derivations and Side Notes

A.2 Radial Acceleration of Platform Junction Points in Spherical Coordinates

The spherical basis vectors are tangential to the coordinate lines and form an orthonormal basis 𝑒r, 𝑒α, 𝑒β that is dependent on the current position 𝐩. Compare Figure A-1, which illustrates the spherical

basis at point 𝐩.

𝑒r = cos 𝛼 sin 𝛽 𝑒x + sin 𝛼 sin 𝛽 𝑒y + cos 𝛽 𝑒z (A-9)

𝑒α = −sin 𝛼 𝑒x + cos 𝛼 𝑒y (A-10)

𝑒β = cos 𝛼 cos 𝛽 𝑒x + sin 𝛼 cos 𝛽 𝑒y − sin 𝛽 𝑒z (A-11)

The angular velocity of the spherical basis is given by:

𝜔 = ��𝑒β + ��𝑒z

(A-12) = �� cos 𝛽 𝑒r + ��𝑒α − �� sin 𝛽 𝑒β

The time derivatives of the spherical basis vector are given by:

��r = 𝜔 × 𝑒r = �� cos 𝛽 𝑒r × 𝑒r + ��𝑒α × 𝑒r − �� sin 𝛽 𝑒β × 𝑒r

(A-13) = ��𝑒β + �� sin 𝛽 𝑒α

��α = 𝜔 × 𝑒α = �� cos 𝛽 𝑒r × 𝑒α + ��𝑒α × 𝑒α − �� sin 𝛽 𝑒β × 𝑒α

(A-14) = −�� cos 𝛽 𝑒β − �� sin 𝛽 𝑒r

��β = 𝜔 × 𝑒β = �� cos 𝛽 𝑒r × 𝑒β + ��𝑒α × 𝑒β − �� sin 𝛽 𝑒β × 𝑒β

(A-15) = �� cos 𝛽 𝑒α − ��𝑒r

The time varying point 𝐩 has position vector 𝐫 in cartesian coordinates, velocity �� and acceleration ��, which can be computed from spherical coordinates as follows:

𝐫 = 𝑟𝐞𝑟 (A-16)

�� = ��𝐞𝑟 + 𝑟�� sin 𝛽 𝐞α + 𝑟��𝐞β (A-17)

�� = (�� − ��2 sin2 𝛽 − 𝑟��2)𝐞𝑟 + (𝑟�� sin 𝛽 + 2�� sin 𝛽 + 2𝑟�� cos 𝛽)𝐞α +

(𝑟�� + 2�� − 𝑟��2 sin 𝛽 cos 𝛽)𝐞β (A-18)

Thus, the radial component of �� is:

rr = (�� − ��2 sin2 𝛽 − 𝑟��2) (A-19)

Appendix A: Derivations and Side Notes 101

xLi

zLi

bi

pi

βi

αi

yLi

eβ er

eα

r

Figure A-1: Spherical coordinates at point p

A.3 LoS Reconstruction

The LoS is defined by two points, the detector center point and the MAM node. Its direction is given by the normalized vector:

𝐜D P,0 = 𝐜D P

‖ 𝐜D P‖

(A-20)

The nominal LoS direction is defined along the z-axis of the detector frame. Thus, the true LoS orientation can be defined by two consecutive rotations around the x- and y-axes of the detector frame:

𝐑LoSD = 𝑟𝑜𝑡y(𝜃)𝑟𝑜𝑡x(𝜙) =

[𝑐(𝜃) 0 −𝑠(𝜃)0 1 0𝑠(𝜃) 0 𝑐(𝜃)

] [

1 0 00 𝑐(𝜙) 𝑠(𝜙)

0 −𝑠(𝜙) 𝑐(𝜙)]

(A-21)

=

[

𝑐(𝜃) 𝑠(𝜃)𝑠(𝜙) −𝑠(𝜃)𝑐(𝜙)

0 𝑐(𝜙) 𝑠(𝜙)

𝑠(𝜃) −𝑐(𝜃)𝑠(𝜙) 𝑐(𝜃)𝑐(𝜙)]

This rotation matrix can be reconstructed from a known position of the MAM relative to the detector,

i.e. 𝐜D P, as follows:

𝐜D P,0|D = 𝐑LoS

DT [001]

(A-22)

[

𝑐D P,0|D,x

𝑐D P,0|D,y

𝑐D P,0|D,z

] = [

𝑐(𝜃) 0 𝑠(𝜃)

𝑠(𝜃)𝑠(𝜙) 𝑐(𝜙) −𝑐(𝜃)𝑠(𝜙)

−𝑠(𝜃)𝑐(𝜙) 𝑠(𝜙) 𝑐(𝜃)𝑐(𝜙)] [001] = [

𝑠(𝜃)

−𝑐(𝜃)𝑠(𝜙)

𝑐(𝜃)𝑐(𝜙)]

From line one and two of Eq. (A-22), the rotation angles can be reconstructed as follows:

𝜃 = asin( 𝑐D P,0|D,x) (A-23)


𝜙 = asin (−

𝑐D P,0|D,x

𝑐(𝜃))

(A-24)

Next, the effect of small changes in MAM position due to thermal distortions of the telescope on the LoS orientation is analyzed. Let the normalized vector including thermal distortions be defined as:

𝐜D Pth,0 =

𝐜D Pth

‖ 𝐜D Pth‖

(A-25)

with

𝐜D Pth = 𝐜D P + [

𝛿th,x𝛿th,y𝛿th,z

]

Comparing this to Eq. (A-22) gives:

[

𝑠(𝜃)

−𝑐(𝜃)𝑠(𝜙)

𝑐(𝜃)𝑐(𝜙)] =

1

‖ 𝐜D Pth‖[

𝑐D P,0|D,x + 𝛿th,x

𝑐D P,0|D,y + 𝛿th,y

𝑐D P,0|D,z + 𝛿th,z

]

(A-26)

Note that the LoS orientation change due to thermal distortions is expected to be very small (in the range of arcseconds) and thus only needs to be taken into consideration for fine pointing after the hexapod maneuver. In this case, the angles 𝜙 and 𝜃 are small, too, and thus the small angle approximations sin (𝛿) ≈ 𝛿 and cos (𝛿) ≈ 0 for 𝛿 ≪ 1 can be applied to Eq. (A-26) leading to:

𝜃 =

𝑐D P,0|D,x+𝛿th,x

‖ 𝐜D Pth‖

(A-27)

𝜙 =

𝑐D P,0|D,y+𝛿th,y

‖ 𝐜D Pth‖

(A-28)

With that the LoS orientation matrix becomes:

𝐑LoSD ≈

[ 1

( 𝑐D P,0|D,x+𝛿th,x)( 𝑐D P,0|D,y+𝛿th,y)

‖ 𝐜D Pth‖2 −


‖ 𝐜D Pth‖

0 1 −𝑐D P,0|D,y+𝛿th,y

‖ 𝐜D Pth‖


‖ 𝐜D Pth‖

𝑐D P,0|D,y+𝛿th,y

‖ 𝐜D Pth‖

1]

(A-29)

With 𝑐D P,0|D,x, 𝑐D P,0|D,y, 𝛿th,x and 𝛿th,y being small in comparison to 𝑐D P,0|D,z and thus ‖ 𝐜D Pth‖, it is:

( 𝑐D P,0|D,x+𝛿th,x)( 𝑐D P,0|D,y+𝛿th,y)

‖ 𝐜D Pth‖2 ≈ 0

(A-30)

and thus Eq. (A-29) can be written as:

Appendix A: Derivations and Side Notes 103

𝐑LoSD ≈

[ 1 0 −

𝑐D P,0|D,x

‖ 𝐜D Pth‖

0 1 −𝑐D P,0|D,y

‖ 𝐜D Pth‖

𝑐D P,0|D,x

‖ 𝐜D Pth‖

𝑐D P,0|D,y

‖ 𝐜D Pth‖

1]

+

[ 0 0 −

𝛿th,x

‖ 𝐜D Pth‖

0 0 −𝛿th,y

‖ 𝐜D Pth‖

𝛿th,x

‖ 𝐜D Pth‖

𝛿th,y

‖ 𝐜D Pth‖

0]

(A-31)

= 𝕀3x3 +1

‖ 𝐜D Pth‖([

𝑐D P,0|D,y

− 𝑐D P,0|D,x

0

]

x

+ [

𝛿th,y𝛿th,x0

]

x

)

A.4 Bang Slew Bang Trajectory

The so-called bang-slew-bang trajectory is characterized by a constant acceleration phase, a constant speed phase and finally a constant deceleration phase as illustrated in Figure A-2 (a).

(a)

t

ω

t

ω

t

ϴ

(b)

t

ω

t

ω

t

ϴ

Figure A-2: Acceleration, velocity and distance plots over time for (a) bang-slew-bang and (b) bang-bang maneuver

For given maximum acceleration 𝑎max, maximum velocity 𝑣max and distance travelled during the maneuver 𝑠end, the duration of the acceleration and deceleration phase 𝑡acc as well as the duration of the constant speed phase 𝑡slew must be computed. Note that the duration of the constant speed phase may be zero, if the maximum velocity is not reached during the maneuver as illustrated in Figure A-2 (b). The maneuver is then also called bang-bang maneuver. The distance travelled during the acceleration and deceleration phase is thus computed as follows:

𝑠acc/dec = min (𝑠end,𝑣max2

𝑎max)

(A-32)

With 𝑠acc/dec = 2(1

2𝑎max𝑡acc

2 ) and Eq. (A-32), the acceleration phase duration 𝑡acc can be computed

as follows:

𝑡acc = √𝑠acc/dec

𝑎max

(A-33)

The slew duration is given by:


𝑡slew = 𝑠end−𝑠acc/dec

𝑣max

(A-34)

If instead the acceleration phase duration 𝑡acc, the slew duration 𝑡slew and the total travel distance 𝑠end are given, the required acceleration for this trajectory can be computed as follows. The total travel distance is defined by:

𝑠end = 2 (1

2𝑎max𝑡acc

2 ) + 𝑣max𝑡slew

(A-35) = 𝑎𝑡acc2 + 𝑎𝑡acc𝑡slew

Note that 𝑣max in this case is not the maximum allowed velocity but the maximum velocity reached during the maneuver with 𝑣max = 𝑎𝑡acc. From Eq. (A-35) the required acceleration for the maneuver can be computed as follows:

𝑎 = 𝑠end

𝑡acc2 +𝑡acc𝑡slew

(A-36)

If instead the maximum acceleration 𝑎max, the total maneuver time 𝑡man = 2𝑡acc + 𝑡slew and the total travel distance 𝑠end are given, the acceleration phase duration 𝑡acc and slew duration 𝑡slew can be computed as follows. Once the acceleration phase duration is known, the slew duration can be computed as:

𝑡slew = 𝑡man − 2𝑡acc (A-37)

The total travel distance can be expressed in terms of maximum acceleration, acceleration phase duration and total maneuver time as follows:

𝑠end = 𝑎max𝑡acc2 + 𝑎max𝑡acc𝑡slew

(A-38) = −𝑎max𝑡acc2 + 𝑎max𝑡acc𝑡man

Solving Eq. (A-38) for 𝑡acc leads to:

𝑡acc = 𝑎max𝑡man∓√𝑎max

2 𝑡man2 −4𝑎max𝑠end

2𝑎max

(A-39)

For 𝑎max > 0 it is √𝑎max2 𝑡man

2 − 4𝑎max𝑠end < 𝑎max𝑡man and thus both solutions of Eq. (A-39) are positive. It is usually beneficial to choose the smaller solution resulting in a longer acceleration phase with lower acceleration rate.

105

Appendix B Simulation Results

Simulation Results

B.1 Simulation Case 1.1: Nominal Operational Flow + Classical Re-Pointing

Note that for the current simulations, the attitude controller has been in steady state mode during the hexapod maneuver instead of idle mode as anticipated. This is due to the fact that the current implementation requires the SC guidance to be computed prior to the simulation. The controller error when activating the attitude controller after the hexapod maneuver then leads to long re-settling times. Therefore, it is planned to adjust the implementation such that the SC guidance is computed online during the simulation based on the actual attitude after the hexapod maneuver. These changes are expected to have a very small impact on the overall performance assessment as the SC attitude changes due to the hexapod maneuver are very small compared to the actual slew. However, major changes to the attitude controller or its implementation have been out of the scope of this thesis.

SC

ACS Mode

Target Slew

idle slew

Hex.

Instrument Switch +

FocusLoS Corr.

Figure B-1: SC boresight over time for simulation case 1.1

106 Appendix B: Simulation Results

B.2 Simulation Case 1.2: Enhanced Operational Flow + Classical Re-Pointing

Instrument Switch +Focus

SC

ACS Mode

Target Slew

slew

Hex. LoS Corr.


B.3 Simulation Case 2.1: Nominal Operational Flow + Enhanced Re-Pointing

Note that for the current simulations, the attitude controller has been in steady state mode during the hexapod maneuver instead of idle mode as anticipated as already described in Chapter B.1.

Appendix B: Simulation Results 107

SC

ACS Mode

Target Slew

idle slew

Hex.

Instrument Switch +

FocusLoS Corr.


B.4 Simulation Case 2.2: Enhanced Operational Flow + Enhanced Re-Pointing

Instrument Switch +Focus

SC

ACS Mode

Target Slew

slew

Hex. LoS Corr.


109

Glossary

Active movement is a term used to describe the motion of a body that is actively controlled with the

help of actuators.

Corrective maneuver is a term used to describe the maneuver performed by the hexapod to

compensate thermal distortions of the telescope structure.

Focus adjustment is a term used to describe the distance along the line of sight between the selected

detector and the mirror assembly module.

Hexapod base is the fixed part of the hexapod mechanism relative to which the platform moves.

Hexapod is a term used to describe a parallel mechanism with six linear actuators that connect a fixed

base plate and a movable platform. The movable platform can thus be controlled in six degrees of

freedom. Such a mechanism is also known as Gough-Stewart platform.

Hexapod platform is the movable part of the hexapod mechanism that is connected to the fixed

hexapod base via six linear actuators.

Hexapod pose is another term for hexapod state.

Hexapod state is defined by the three lateral degrees of freedom of the hexapod platform (x, y and z-

offset relative to a nominal position) and the three rotational degrees of freedom (roll, pitch and

yaw angle of the platform).

Instrument switch is a term used to describe the maneuver performed by the hexapod to point the x-

ray beam towards a different instrument.

Instrument Switch Mechanism is equivalent to the hexapod mechanism here.

Line of Sight is the line defined by the center point of a detector and the nodal point of the mirror

assembly module and defines the direction into which telescope is ‘looking’.

Mirror Optical Axis is the line through the nodal point of the mirror assembly module, perpendicular

to the center plane of the mirror assembly module.

MOA misalignment is a term used to describe the error angle between the line of sight and the mirror

optical axis.

Operational Flow is a term used to describe the chronological order of the different hexapod and

spacecraft maneuvers performed between two observations.

Passive movement is a term used to describe the motion of a body that is caused by the motion of

another body to which it has some form of mechanical connection.

Spacecraft main body is a term used to describe the spacecraft body without the hexapod mechanism

and the mirror assembly module.

Sun exclusion zone is a half cone arround the sun direction (straigt line from spacecraft to sun), which

must not be entered by the telescope boresight to prevent sunlight from falling into the telescope.

110 Glossary

Tait-Bryan angles describe the angles for a sequence of rotations (e.g. x-y-z sequene) usually used to

describe the orientation of aircrafts and spacecrafts relative to a world frame. Often mixed up with

Euler angles (e.g. x-y-x sequence)

Target line is the straight line between the spacecraft and the observation target.

111

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